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ON THE PHYSICS OF PERCEPTION
Willy Wong
A thesis submitted in conformity with the requirements
for the degree of DOCTOR OF PHILOSOPHY Graduate Department of Physics and Institute of Biomedical Engineering
University of Toronto
@ Copyright by Willy Wong 1997
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On the Physics of Perception PhD 1997
Willy Wong, Department of Physics and hstitute for Biomedical Engineering University of Toronto
Abst ract
A single equation has been discovered governing the flow of information or entropy
fiom a sensory stimulus (such as light or sound) to the organ of sensation (such as a
rod or a hair cell) which can account for nearly all expenmental results recorded in
sensory neurophysiology relating stimulus intensity and stimulus duration to biolog-
ical response. This equation is capable of handling any time-varying single-stimulus
input. The biological response is taken as the frequency of neural discharge.
When this equation is extended to the psychophysical level, the theory is capable
of unifymg many phenornena associated with thresholds.
There may be no other sensory theory of comparable generality. The scope and
limitations of the theory are discussed.
Acknowledgment s
1 would like to take this opportunity to thank some of the people who have helped
make t his t hesis possible.
First of dl, I a m greatly indebted to my supervisor, Prof. Ken Norwich, for his
contribution to this work. This thesis represents a coilaborative effort between Prof.
Nornrich and myself.
1 would like to thank my supervisory committee, Prof. Rashmi Desai, Prof. Sajeev
.John and Prof. H a m Kunov for their many helpfiil suggestions, criticisms and com-
ments regarding the research. This thesis would not be possible without their insight
and expertise.
In particular, 1 would like to acknowledge the generosity of Prof. Kuuov for grant-
ing me the use of all materials and equiprnent in the Bioacoustics Laboratory at the
Ins titute of Biomedical Engineering. His generosity is great ly appreciat ed.
A special word of thanks to Prof. Alf Dolan, Prof. bfichael Menzinger (chemistry)
and Prof. John Perz, members of my final examination committee. 1 am particularly
gratefd to Prof. Perz for his methodical review of my thesis.
1 woiild also like to thank Prof. Lawrence Wmd af the University of Bntish
Colimbia for his many helpfd suggestions and comments as extemal reviewer.
This research was supported by fellowships fiom the University of Toronto and the
... Ill
Governrnent of Ontario. Furthermore, much of the research was carried out with the
aid of an NSERC operating g a n t to Prof. Ken Norwich.
Words cannot elcpress the gratitude that 1 owe to my parents? Bing and James,
and Auntie Helen for al1 their support and encouragement. 1 couldn't have gone this
far withoiit you!
To my lovely Etsuko ... thank you so much for helping me grow iip, for showhg
me that there is more to Me than just intellectual pursuits.
And h d y , to Ken Norwich, a friend and teacher for nearly 7 years, you have
inspired me in a way like no other scientist. 1 will forever remember your limitless
creativity, honesty, generosity and patience.
To yoii, Ken, 1 dedicate this thesis.
Contents
1 Introduction 1
2 An Overview of Sensory Physiology and Psychology 5
1 The Sensory Systern.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . - . . . . 5
2 The Physiology of the Senses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1. I Neira1 Adaptation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.1.1 Early Rise in the Adaptation C w e . . - 8
2.1.2 Spontaneous Activity . . . 10
2.1.3 De-adaptation . . . 10
2.2 Driven Neural Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 Empirical Laws Goveniing the Neural Response . . . . . . . . . . . . . . . . . . . . . 1 1
3 The Psychology of the Senses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.1 Psychophysical Adaptation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.2 Magnitude Estimation and the Law of Sensation.. . . . . . . . . . . . . . . . . . . 15
3.3 Threshold Phenornena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.3.1 Absolute Threshold . . . 18
3.3.2 Differential Threshold . . . 22
3.4 Simple Reaction Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26
.............................................................. 4 AFinalNote 31
5 Conclusions .............................................................. - 3 2
3 The Sensory Entropy Theory 34
1 Classical Entropy Theory (pre-1993) ...................................... 35
1.1 Properties of the Classical Entropy Equation ....................... - 3 7
1.2 Failures of the Classical Theory .................................... - 4 0
2 hiversal Entropy Theory (1993-) ......................................... 41
2.1 Deriving the Universal Entropy Equation .......................... - 4 1
2.2 Siimmary ............................................................ 48
4 Neural Explorations with the Universal Entropy Equation 50
1 Neural Adaptation ....................................................... -50
2 Spontaneoiis Neiiral Activity .............................................. 51
3 Early Rise in the Adaptation Cime ....................................... 54
3.1 Correlated Receptor Samples ........................................ 54
3.2 Effects of Stimulus Rise Tirne on Adaptation ....................... - 5 5
..................................................... 4 NelualDe-adaptation 56
5 On the Relationship Between the Classical and the Universal
Entropy Equation ......................................................... 58
6 Finther Experimental Validation of the Universal Entropy Equation ..... 58
6.1 Ailditory Neural Response ........................................... 59
vi
6.2 O t her Cases ........................................................ -60
7 Other Investigators ........................................................ 63
5 Uni&ing Threshold P henomena 64
1 Thresholds and the Neiiral Response ..................................... - 6 6
3 Introdiicing the Threshold Hypothesis .................................... 67
3 Differential Thresholds and Weber Fractions .............................. 71
3.1 Deriving the Differential Threshold Equation ....................... - 7 1
3.2 Implications of the Differential Threshold Equation ................ -72
3.3 Validating the Differential Threshold Equation ..................... - 7 3
3.3.1 Visual Differential Thresholds . . -73
3.3.2 Auditory DifFerential Thresholds ... 76
3.4 A Generalization of the Threshold Hypothesis ...................... 80
4 Absolute Thresholds ..................................................... - 8 1
4.1 Bloch's Law: Part I ................................................ - 8 3
6 Simple Reaction Time 84
1 Deriving Piéron's Law ..................................................... 84
.................. 2 Implications of the Entropic Derivation of Piéron's Law 88
2.1 Bloch's Law: Part II ................................................ -89
7 Speculating on the Psychophysical Response 91
vii
Psychophysical Adaptation ................................................ 92
Magnitude Estimation .................................................... 96
2.1 The Measurements of Riesz .......................................... 96
2.2 Magnitude Estimation ............................................... 98
8 Discussion
................................................... 1 The Interna1 Signal. 61 101
............................... 2 A Common Set of Acoustical Parameters? 104
3 The Threshold Associated with Simple Reaction Time .................. -105
.................................................... 4 Two Remaining Points 106
9 Conclusions 107
1 CreneralRemarks ......................................................... 107
3 Specific Remarks ........ .. ............................................... 108
2.1 Neurophysiologicd Phenornena .................................... -108
2.2 Psychophysical Phenornena ........................................ -109
A 4 Mode1 of Equilibrium Receptor Memory 112
1 Physical Model ........................................................... 112
2 Derivation of the Power Dependence of Receptor blernory ............... 114
B Deriving the Neural Response to a Double Step Input
C Deriving the Differential Threshofd Equation for the Continuous
Increment 121
D Deriving Eq. (59)
E The Absolute Threshold of Human Hearing
F MapIe Worksheets
Bibliography
Chapter
Introduction
The stiidy of the senses is a highly interdisciphary endeavour. A complete stiidy
may involve a wide range of disciplines including mat hemat ics, philosophy, physics,
physiology and psychology.
In this thesis, 1 hope to present a unified and conceptual approach towards the
iuiderstanding of fundamental sensory processes. M y approach is primarily fcom the
point of view of a physicist.
Perhaps it is important to realize what this thesis is not about. The thesis is
not about the detailed rnechanisms of sensory processes. That is, 1 s h d not be
concerned with the functional differences between the eyes and the eus. While siich
a stiidy woidd obviousiy be of importance to the understanding of sensory physiology,
1 shall instead focus on the questions: 'What is sensation?" "What is the nature of
perception?"
To approach issues of such fundamental nature, we cannot concern ourseives with
the details of every sensory organ. Instead, we rnust strive for a coherent view of the
entire sensory system by emphasizing the features common to a l l sense organs. One
common feature is their ability to receive and transmit information. This idea was
first siiggested in 1965 (see Norwich. 1983). Thus, perception is linked to information
or entropy. Entropy becomes, so to speak, the "fundamental currency" of perception.
Central to the entropic approach is the concept of uncertainty. During the per-
ceptiial process, the organism acquires information £rom the stimulus signal through
the process of sampling. This information reduces the "uncertainty" imparted to the
receptor at the time of first contact with the stimulus signal.
By conjecture, the associated sensory response is then equated to this uncertainty.
The greater the stimulus uncertainty, the greater the sensory response.
A mathematical theory embodying these principles have been under development
for the past 30 years. The entropy theory has been put forth as a theory capable of
iuiifying most if not al1 of sensory studies involving the three main variables: sensory
response, stimulus intensity and duration of stimulation.
One important strength of the sensory entropy theory is its ability to predict the
action and response for almost all sensory modalities (vision, audition, gustation,
olfaction, muscle, temperature) in a.ll complex organisms.
The reader might be suspicious of such an extravagant claim and be surpriseci
that the sensory system can even be addressed with a single theory. In this age
of increasing complexity, science has tended to stray fiom its original conceptual
roots. and to approach nature empirically. There are, for example, many competing
approaches which treat the sensory system purely as a non-linear black box of many
parameters (eg. Saglam et al., 1995). While such an approach wodd mode1 every
nitance observed in the biologically measured response, a non-linear black box would
probably not give hirther insight h to the concept of sensory perception.
What the entropy theory lacks in terrns of fine details, it gains in the larger picttire.
In that seme, t h s approach is similar to the style of art known as Impressionism. For
example, several of the Impressionistic artists painted with the technique of pointil-
Lism. While their works lack realisrn when viewed closely, a larger picture emanates
from the camvas when seen farther back.
As with any other theory, the sensory entropy theory can be stripped of al1 its
interpretation Ieaving only a 'Lcalciilator" of sensory variable.. However, just as the
eqillvalence principle is an essential and inseparable feature of generd relativity, we
siiggest that the strength of the entropy theory lies prirnarily in its interpretational
and not mathematical power.
In this thesis, 1 shall use data obtained from many sensory modalities to test
the theory. However, for certain phenomena such as sensory thresholds, complete
sets of experiments can only be found for a fav modalities like audition and vision.
Nevertheless, it is expected that the entropy equations should also hold for data
acquired hum O t her modalities as weU.
This thesis is divided into 9 chapters. In Chapter 2, the basic ideas of sensory
physiology and psycholog- are introduced. This is an important chapter because
al1 of the later developments require this background knowledge. Chapter 3 deals
with the derivation of the ''master equation" within sensory entropy theory. The
follow-ing Chapters 4-7 deal with the application of this equation to the unification
of sensory studies. Chapter 8 reviews some of the new ideas appearing in this thesis,
and Chapter 9 concludes the thesis.
Chapter 2
An Overview of Sensory Physi-
ology and Psychology
In this chapter I introdvce the basic concepts and ideas ofsensory physiology
and psychology relevant to the research.
2.1 The Sensory System
The sensory system is a direct link between the organism and the external world. The
information communicated from the external environment is mediateci by the sensory
stimuli detected by the sense organs. For example, vibrations of air molecules are
translated into the sounds we hear and the energy of photons form the images we see.
The transduction process responsible for changing physical stimuli (eg. sound waves
or photons) into the resulting sensory experience is not hilly imderstood.
The sensory system can be categorized roughly into three main regions. The neural
periphery is the area where the organism first cornes into contact with external stimuli.
These extemal signals are detected by the sensory receptors (eg. the rods or cones
in the eye. the hair cells of the inner ear, etc.). Associated with the r-ceptors are the
primary or 6rst-order afferent neurons w-hich respond to the stimuli detected by the
receptor. The neurons Lire action potentialç with a coded frequency whereby stimuli
of higher intensities elicit higher firing frequencies. and vice versa.
These neilral impulses proceed along the peripheral and central neruous system
towards the oiiter region of the brain known as the sensory cortex. It is believed
that this region is responsible for converting the freqiiency of neural impulses to the
residting cognate experience of sensation (eg., "1 hear a loiid soiuid" or "1 see a bright
light" ) .
The three regions of the sensory system are illustrateci schematically in Fig. 1.
The regions are conveniently labeiled as: A (neiird periphery), B (nervoiis system)
and C (brain).
Responses originating £iom the neural periphery will be termed the neural response.
Responses originating £rom the conscious apprehension of the stimulus will be termed
the psychophysical response. The neural response is typically rneasured in terms of
the freqiiency of nerve impulses generated by the first-order sensory neuron. The
psychophysicd response is generally measured by some subjective scale to rate the
sensory magnitude of stimuli.
It has long been observeci that the neural response is directly related to the psy-
Region A Neural Peripbery
Region B Peripberal and
Central Nervous System
Region C Brain
Sensory Cortex
Sensory Nerve Fibre Nerve Impulses
chophysical response (eg. Borg et al., 1967). Greater neural responses are generally
associated with greater psychophysical responses, and vice versa.
2.2 The Physiology of the Senses
MI of the basic sensory modalities share certain common physiological features.
2.2.1 Neural Adaptation
If the isolated single sensory receptor is excited by stimuli of h e d intensity and
prolonged duration, a characteristic response is observed at the associated afferent
netaon. The freqiiency of nerve impulses is greatest near the initial onset of stimulus7
falling s teadily with prolonged stimulus duration. This phenornenon is termed neural
adaptation and is observed nearly universdy for aJl sensory modalities. For example,
the data of Adrian and Zotterman (1926) are plotted in Fig. 2. They measured the
firing fiequency of impulses in a frog muscle spincile under a load of a 1 g weight.
Adaptation occimeci over the period of 5 to 20 seconds corresponding to the time
when the muscle was loaded.
In addition to the effects of adaptation, other features have &O been observed in
the neiilal responses of the various sensory modalities.
2.2.1.1 Early Rise in the Adaptation Curve
Upon omet of stimulus, the firing frequency requires some tirne before reaching the
Figae 2: Adaptation in a £iog muscle fibre. A 1 g load was applied to the muscle
from time 5 to 20 seconds. Data fiom Adrian and Zotterrnan (1926).
maximiun rate. For example, in Fig. 2' we see that approximately 2 seconds were
reqilired before the muscle nerve fibre reached its maximum £king rate.
2.2.1.2 Spontaneous Activity
Most sensory neiuons actively generate neinal signals even in the absence of an ex-
ternal input. In the case illiistrated in Fig. 2, we see a significant arnount of activity
generated by the muscle spindle prior to the loading.
When an input is terminateci, the neural response describes a mirror image of the
adaptation ciwe (Dudel, 1986). Typically, the response will fall abruptly at the end
of stimulation, and gradudy show a rnonotonic rise towards the spontaneous rate.
De-adaptation is observed From time 20 seconds and onwards in Fig. 2.
2.2.2 Driven Neural Response
Complementary to adaptation is the eEect of the driven neural response. How does
the receptor/neiiron complex respond to stimuli of k e d duration and Mering in-
tensities? When we observe the data of Smith in Fig. 3, we notice that the firing
freqiiency increases wit h increasing stimulus ma,pitude. These data were recorded
in the auditory fibers of the Mongolian Gerbil ear and the stimulus is a pure tone of
diuation 1 ms. Near the high intensities, the neural response saturates because the
fibres are incapable of responding at any greater firing frequencies. At the very low
intensit ies. the response approaches the level of spontaneous activity. Similar charac-
teristics have been obsenred with other investigations as weil (eg. compare Galambos
and Davis. 1943).
In siimmary. we see that the neural response is primarily dependent on the two
lwiables. stimulus intensity and stimulus duration. F9r 6xed intensities. increasing
the dtiration will decrease the firing rate. At fked durations, increasing the
sity will increase the firing rate. This result is summdzed by an idealized
dimensional plot of intensity and diuation versus response in Fig. 4.
inten-
three
2.2.3 Empirical Laws Governing the Neural Response
bIany empirical observations and equations have been formulateci to account for the
experimentdy observed neural response below saturation levels.
In terms of the driven neural response (cf. Fig. 3), many investigators (eg. Stevens.
1970: Schmidt. 1986) have observed that the firing rate F is approximately a power
function of the stimulus intensity I :
where Ci is a constant and n is some constant greater than zero. Other investigators
(eg. Hartline and Graham, 1932) have found that the firing rate obeys a logarithmic
law of the form
Figure 3: The driven neural response of an auditory fibre to sound stimulation. Data
were recorded hom the inner hair cell of a Mongolian Gerbil. From Smith (1988).
Figure 4: Three-dimensional plot showing an idealized neural response to both stim-
idiis intensity and duration: F = F ( I , t ) .
where CI, and C3 are constants.
For adaptation response, most investigators tend to use a mono- or double-exponential
hmction (eg. Bohnenberger, 1981). For example a moneexponential function would
take on the form
where a serves as the adaptation time constant.
I t is important to remember that Eqs. (1), (2) and (3) sre d empirical. They
were al1 obtained in a pimely ad hoc fashion for the purpose of fitting experimental
data.
2.3 The Psychology of the Senses
From the neural periphery, we now tiirn our attention towards the psychophysical
response (region C in Fig. 1). Ever since the discovery of the nerve impulses, it
has been posttilated that the psychophysical response m u t be, in some way, derived
from the neural response. We now examine a number of properties for which the
psychophysical response seems to rnirror the neural response.
2.3.1 Psychophysicd Adaptation
The psychophysical analogue of neural adaptation is something we ail experience on
a daily bais. Any pimgent odour thst catches one's attention will gradually fade
14
from the sensorium with prolonged stimulation. For example, the smeil of paint is
greatest when p i h s t enter a newly painted room.
In Fig. 5, the data of Gent and McBurney for psychogustatorid adaptation are
plotted. Saltiness was measured by the investigators (using a subjective scale) for
varying diuations of stimulation. The stimulus was a salted filter paper placed on the
human tongiie. One again. we notice that sensory response is a monotonic decreasing
fimction of stimulus duration. This same effect has been obsewed for the other
sensory modalities as well.
One very rnarked difference between psychophysical and neural response is the time
course over which adaptation proceeds. For certain rnodalities like vision or aiidition,
neural adaptation is largely complete by about 100 ms, whereas, psychophysical adap-
tation may occiu over a length of several minutes. The reason for the Werence in
time scales for the two processes is still a mystery to researchers. Although both
responses tend to be similar, it is not at all clear how the psychophysical response is
derived from the neural response.
2.3.2 Magnitude Estimation and the Law of Sensation
Corresponding to the driven neural response is the law of sensation. The law of
sensation states t hat psychophysical sensation increaçes as a monotonic function of
increasing stimulus intensity (with duration held constant). Depending on how the
siibjective scale is chosen, the data may be fitted with either a power law championed
Figure 5: Psychogustatory adaptation recordecl for a NaCl stimulus. Data £rom Gent
and hlcBiirney (1978).
by Stevens (1936. 1961).
where @ is the subject's sensory response or magnitude estimation corresponding to
a stimidtis of intensity I , or may follow a logarithmic law as proposed by Fechner
(18601 19661,
in direct analogy with the two empirical neural Eqs. (1) and (2). Indeed, since
many investigators made the identification of !P with F, the Stevens' and Fechnerian
psychophysicd laws provided the mode1 for both neural Eqs. (1) and (2).
While the logarithmic law was originally proposed by Fechner, Stevens was not the
first person to stiggest the power law. The honour goes to a Belgian physicist narned
J. A. F. Plateau who worked in the nineteenth century. Interestingly, Fechner was
also a physicist, working originally on voltaic cells in the mid-1800's. Fechner, who
coined the term psychophysik, is now known as the father of psychophysics.
The first person to apply the power law to psychoacoustics was the physicist H.
Fletcher (Fletcher and Mtmon, 1933). Before comrnencing his studies on human
hearing at the AT&T Bell Labs, Fletcher was an assistant to Robert Millikan in the
famoiis oil drop experiments.
RLiich of the data from experiments on magnitude estimation do not conform ac-
ciuately to either the log or the power law. Consider for example the taste data for
17
NaCl of Stevens (1969) plocted in Fig. 6. Neither fùnction will fit the data prop
erly over the entire range. We see that there is a tendency for the lower data points
to conform to a power function, while the higher points conform more closely to a
logarit hmic fiuiction. This tendency has been observed with quite a few other experi-
ments and woidd tend to suggest that there is another more complete psychophysical
law encompassing both Eqs. (4) and (5) as the limiting cases for both s m d and large
stimidirs intensities.
2.3.3 Threshold Phenornena
Al1 of the psychophysical phenomena we have considered thus far have had clear neural
analogues. That is, psychophysical adaptation is the analogue of neural adaptation,
and the law of sensation is the analogue of the driven neural response. We now t i m
our attention to another class of phenomena which does not have, or does not appear
to have, a clear neural analogue.
2.3.3.1 Absolute Threshold
In his book What is Life?, Schrodinger speculates on the reasons why the brain,
with a "sensorial system" attached to it, is not sensitive to single atoms. Since the
brain is capable of very orderly and coherent thought processes, he reasons that the
physical stimuli (atoms, molecules, etc.) which account for the perceptual processes
miist also obey strict physical laws. For, if the random "atomistic" fluctuations were
Figure 6: Magnitude estimation for NaCl solution. Data from Stevens (1969). The
open circles correspond to data plotted with the logarithmic left axis and the H.led
circles correspond to the h e a r right &S.
perceived directly. the information transmitted to the brain would not resiilt in any
orderly percept or thought process. However, since a large ensemble of atoms wiU
always obey statistical laws, he postdates that the o d y way the brain is capable of
maint aining well-ordered t hought processes is by perceiving only stimuli of s a c i e n t
intensity which will obey strict physical laws to a high degree of accuracy.
We see that Schrodinger is duding to the idea of a sensory threshold below which
no stimulus is detectable. We terni this the absolute detection threshold. The absoiute
threshold is always d e h e d with some statistical criterion in mind because the sensory
system itself is siibject to fluctuations on a moment to moment basis. Hence, we may
define the absolute threshold to be the stimulus intensity which is detectable Say 50%
of the time. Obviously, a m e r e n t criterion will change the absolute threshold. A
90% criterion ~vill give a much higher threshold than a 75% criterion.
Here is one example illustrating the absolute threshold. In 1942, Hecht' Pirenne
and Schlaer determined that rod photoreceptors are sensitive to single photons in
the dark. However, a human subject requires on average 10-20 photons before he or
she can detect the light stimulus 60% of the tirne. Thus, a neural response does not
necessarily imply a psychophysical response.
It is apparent that threshold intensity and stimulus duration must obey some
sort of reciprocal relationship. For example, a weaker intensity may be detectable
provided that it is on for a longer period of time. Thus, Bloch, in 1885, formidated
fis celebrated empirical law relating the absolute threshold intensity. AI, and the
stimulus dination, At:
where C8 is some constant independent of intensity.
However. it soon became apparent that Eq. ( 6 ) could not account for all the
rneasiired data. In audition. Garner (1947) proposed a variant of Bloch's law,
where B is some positive exponent. For siibstantidy lower values for AI, Blondel
and Rey independently proposed an alternative equation for the visual threshold,
where ü is called the Blondel-Rey constant. AIm is the lowest possible threshold given
a Cived statistical criteria. That is, in the limit where the duration of stimulation
becornes very long (At -+ m), A l approaches AI,. An identical equation was also
proposed for aiidition by Hughes (1946).
Eq. (8) has the additional benefit that, for short durations (At < a), it approxi-
mates Bloch's law,
Later researchers were eager to r e h e the Blondel-Rey law because they had found
21
that. the data did not conform too closely to Eq. (8). Instead. it was proposed that
be iised as an alternative expression for the absolute threshold. The constant n serves
miich the same rôle as the inverse Blondel-Rey constant 1/a. Eq. (10) also reduces
to Bloch's law in the iimit of s m d At.
2.3.3.2 Differentid Threshold
So farl we have considered thresholds for absolute detectability. When we are ai
stimulus intensities far above threshold, o w experience teils iis that we perceive in a
continuun. However, is this a red or apparent continuum?
Since we can never jiidge the difference in intensity between two stimuli Mth
arbitrary acciuacy, there must also be a threshold associated with the process cf
differentiation. CVe c d this minimum perceptible difference the dzfferential threshold
AI.
To give an example, we might wish to measure the Merential threshold of heavi-
ness for a hiunan siibject. Given an object of, say, 100 g, how much must the weight
be changed before a just noticeable difference (jnd) in sensation is detected, Say, 75%
of the time?
Siich experiments were carried forth by investigators as far back as 1830's. E.H.
Weber, a physiologist, postulated that if a stimulus of intensity 1 is presented to a
siibject, the fiactional change in intensity to provide a jnd (just noticeable difference)
22
in sensation is constant. That is.
which is known as Weber's law. The expression A I / I is calleci the Weber fraction.
The Weber fraction is the differential threshold divideci by the bzse intensity
From his work. the physicist Fechner was astute enough to make the following
association. Denoting the jnd in sensation as A@, Fechner assimeci that
That is. the jnd in sensation is independent of the magnitude of the intensity. Corn-
bini~ig the two eqiiations, Fechner obtained
which he then integrated by replacing the finite différences with differential quantities
to obtain his famous logarithmic psychophysical law (cf. Eq. (5))
This ~echnique is now c d e d Feclinerian integration.
Returning to E,q. (1 1), we see that Weber's law predicts a horizontal line when
AI/ I is plotted against I. However, later investigation revealed that the experimen-
t d y ob tained data tended to deviate substantially kom Weber's law , particularly
23
for the lower intensities. Typicdy the Weber fraction wodd rise siibstantially for
decreasing values of 1.
Konig and Brodhtm (1889) for vision, and bucisen (1923) and Riesz (1928) for
atidition. obt ained resiilts which tended to conform to the equation
where, for large values of I , the Weber fkaction approaches a constant as predicted
by Weber's Iaw.
Fig. 7 shows Riesz's empirical fits to his own measurements. He perfomed mea-
siirements at several frequencies of sound.
In the limit where I + O, we see that the differential threshold eqiials the absoliite
threshold. For a fixecl statistical criterion, we know that the absolute threshold is
constant. Thus,
Dividing both sides by I we obtain
log (Y) = - log (1) + log (Cii) .
That is. the Weber fraction data when plotted in full logarithmic coordinates (log (A l I I )
vs. log(1)) miist fall with dope - 1 for 1 -+ O. This relationship was observed exper-
imentally by Miller (1947) for audition.
24
Figtve 7: The data of Riesz (1928) showing Weber fraction versus sound intensity
across different Erequencies.
There are many ways to measure the Weber fraction (see Comsweet and Pinsker.
1964: Viemeister, 1988). One method is shown in Fig. 8. This method requires the
siibject to detect an increment (or decrement) over a pedestal of intensity I . Usudy
the pedestal is of a length much longer than the increment. Typicdy, a siibject is
given two intervals with the pedestal. However, only one of the two intervals contains
the increment. The siibject tries to identiS the intend in which the increment was
present. This method of rneasuring the Weber haction is known as the continuous
rnethod.
As with the absoliite threshold, one might expect that the differential threshold
woidd also vary a i th the duration of the increment intensity. For example, retuming
to the continuoits method in Fig. 8, how would the differential threshold change if
we varied At? One might expect that the threshold decreases with increasing At
in a manner similar to the absolute threshold. For example, shown in Fig. 9 are
two cimes measiued by Garner and Miller (1947) showing how the Weber fraction
(or differential threshold) varies with incrernent duration a t two different intensities.
The cimes fall monotonically in agreement with the various empirical laws for the
absolute threshold.
2.3.4 Simple Reaction Time
All of the threshold phenomena considered in the last section showed a reciprocal
relationship between stimulus duration and intensity. We now consider an entirely
Figure 8: Stimulus input for continuous increment.
Figure 9: Weber fraction versus increment duration for audition as measured by
Garner and Miller (1947). The open circles are at 40 dB SL and the iilled circles at
70 dB SL.
different class of sensory phenomena which also exhibits the same relationship.
Simple reaction time mas be defineci as the duration of time that elapses between
stimidus omet and time of first motor response to that stimulus. For example, a
siibject might be required to press a button imrnediately upon detecting a visiial
stimidus. From experience, one knows that a stronger stimulus elicits a faster reaction
time, and a weaker stimulus elicits a sfower reaction tirne.
CVhereas the threshold valiies of A I considered in the last section are always s m d
compared to the fidl physiologicd range of sensation (for example, the differential
threshold? AI, may be less than 1 dB for hearing, compared to the hill range of human
hearing which spans 120+ dB), the valiies of A I for simple reaction are nominal (eg.
see Fig. 10, x-axis).
In Fig. 10. the auditory data of Chochoile (1940) illustrate the relationship between
AI? the stimulus intensity, and At,, the reaction time. The curve shows a monotonic
fa11 of reaction time with increasing intensity. Typicdy, At, does not fall beyond
some minimum which is called, appropriately, At, ,in.
In 1945, the eminent physiologist Piéron discovered that most if not ail of the
simple reaction data measireci fkom the various sensory modalities could be fitted
wi t h the single empirical equation
b 1 8 At, = At,,, + - AIn '
where the exponent n is in approximate agreement with the value obtained bom Eq.
29
Figue 10: Simple reaction time to a 1000 Hz auditory tone as measured by Chocholle
(1940).
( 15) for the same sensory modality. Along with the conctvlence of exponents, the
sirnilarity in form of these two equations is striking (eg. see Ward and Davidson,
1993).
It shoiild be noted that we have adopteci a Merent -bol for the stimuliis dura-
tion in this section. There is a very simple reason for this change. At, is a reaction
time which combines both a sensory and a rnotor response time. It might be helpfid
to think of the detection time, At, as behg purely the time reqillred for the sensory
system to process the signal, and At, is piuely the thne required for the subject to
make any motor response. Thus, it is hypothesized that
At, = At + At,, (19)
an idea which was discussed by Halpern (1986) (see also Norwich, 1993). We rniist
be mindfiil of the differences between At and At, as we proceed with the theoretical
development .
2.4 A Final Note
This section is a brief digression on the sociology (and not the science) of sensory
perception.
From the concept of psychophysical response alone, we can easily see that al1
experiments involving a psychophysical element must invariably be performed on
human siib j ec t s (however , see EvI arks, 1974, "Animal psychophysics" ) .
31
On the other hand, d data obtained fiom the neural periphery are from non-
hiunan siib jects. Because of the restrictions on human e-xperimentation few sen-
sorineiiral experiments have ever been performed on human siibjects.
Quite apart fiom the ethical considerations of whether or not such research should
be carried out on hiunan siibjects, it is important to realize that a liucury not available
to 11s is a cornpiete set of sensory data (both neural and psychophysicd) from a single
organism. For example. there is no chance to compare, Say, neural sensory parameters
wïth psychophysical parameters.
Nevertheiess, we shall try our best to infer the missing experiments from the results
ciirrently available to r i s .
2.5 Conclusions
This chapter has been quite long and involved. However, a proper understanding of
this chapter will give the reader a better appreciation of the developments to follow.
Highlighted here are some of the main points of this chapter:
0 The neural response, F, originates from Region A (cf. Fig. 1).
O The psychophysical response, Q, originates from Region C.
rn The nelirai response can be measured in terms of the hing rate of nerve impulses from the primary aEerent neurons associated with the sensory receptor or detector .
rn The psychophysical response is measured in terms of a subjective scale.
The psychophysical response, to a large extent, mirrors the neural response. al- thoiigh it is not clear how one is denved from the other.
Sensory response decreases with adaptation.
Sensory response increases with increasing stimulus intensities.
The absolute threshold rneasures the minimum perceptible intensity.
The differential threshold rneasures the minimum increment intensity before a change in sensation is perceptible.
Simple reaction time rneasures the time reqiiired to make a motor response given a sensory ciie.
Sensory processing time is different hom reaction time.
There are many cornmon properties shared among the basic sensory modaiities.
The sensory sciences are replete with ernpirical equations.
Chapter 3
The Sensory Entropy Theory
In the last chapter, many of the empirical findings and observations of sen-
sory perception were discussed. No single theory has ever been successful at
unifging even a feu empirical r e sd t s for a single modality. Howeuer, the sen-
so ry entropic approach, under development for the past 30 years, has been put
forth as a theory capable of unifying most of the basic sensory obseruations
within a cornmon mathematical and philosophical frBmevork.
The development of the Sensory Entropy T h e o y can be dzuided into tuio
phases: the classical theory, which 2s the more restricted foremnner, and
the ,universal theory, which is more general and i s capable of even greater
predictiue pouler, although at the ezpense of greater mathematical complexity.
The task o j this chapter is to derive an entropy equation which then sentes
as a master equation or "equation of state". Having obtained the master
equation, i t is t hen a relatively simple task tu manzpulate this equation in
uarious ways to derive most if not al1 of the empirical results presented zn
the last chapter. The application and manipulation of the entropy equation
is the content of the following chapters.
This chapter zs dzvided into two sections. The first part deals with deriving
the classical entropy equation. No attempt is made at being ràgorous. Instead,
1 present a hevristic derivation, concentrating on what makes "sense". That
way, the reader can obtain a better feel for the entropy equation i n its sim,pler
f o m .
The second part deals wzth a full derivation of the universal entropy equa-
tion. At the end of this chapter, we will drop the classical equation and keep
only the universal equation for the remaining chapters.
3.1 Classical Entropy Theory (pre-1993)
In the previous chapter, we designated the psychophysical response by 8. When we
consider hearing, for example, we know that \k m u t be approximately a logarithmic
fimction of intensity because a 40 dB tone sounds roughly twice as loud as a 20 dB
tone. Thlis,
Sepwating these constants gives
where K', 3' and n are all positive constants.
Recdi that 9 decreases with increasing stimulus duration. Thus we might have
althoiigh the iuiits of D' m u t be changed to reflect the change in dimensionality. As
t + oo, all sensation should vanish due to adaptation. Instead, Eq. (23) diverges.
Hence we might modify this equation to obtain
so that as t -, oo, iI, -, 0.
Xssiiming that the psychophysical response exactly rnirrors the ne-ara1 response
jexcept for the difference in time scales) , we can now write
where K is a new constant and ,a < to arcount for the tirne dilation as we proceed
from neural to psychophysical response.
Mthough the connection to entropy or information theory is obscure in this heuris-
tic derivation of Eq. (25), we s h d elaborate fully on the entropic heritage in the
derivation of the universal entropy equation.
36
Eq. (25)' the equation for neural response. was first derived by Norwich in 1977.
This equation was later extended to work at the psychophysical level. That is. Eq.
(25) historically preceded Eq. (24).
A modification of his original derivation from 1977 wiil be presented in Section 3.2.
3.1.1 Properties of the Classical Entropy Equation
As mentioned earlier, the entropy Eqs. (24) and (25) have enjoyed some success at
conferring a degree of unification in the sensory sciences. The ability of the entropy
eqiiations to integrate some twenty or more sensory empirical equations, phenornena'
etc.. is well dociunented in a recent monograph (Norwich, 1993).
A qiuck check of Eqs. (24) and (25) yields the follonring results.
If I is held constant, increasing the stimulus duration will produce residts consistent
with adaptation (although, in this ad hoc derivation, we "built in" the effects of
adaptation, the original derivation did not contain such arbitrary measures). Please
see Norwich and McConville (1991).
When the stimulus duration is heid constant, we can set y = f l / t to obtain
Wien y ln << 1 (stimulus is weak) , taking a first order series expansion, we obtain
in agreement with power law of sensation (cf. Eq. (4)). When yIn » 1 (stimulus is
st rong ) . we have. approximately
* = C** log I + C23,
in accordance with Fechner's logarithmic law. A similar process with Eq. (25) d
yield Eqs. (1) and (2) for the neural response.
We are now in a position to imderstand why neither the power nor the log law cotdd
fit the data in Fig. 6 over the entire range. Since Eq. (26) is the more "complete"
law. ernbracing the two other psychophysicd laws, we can expect a better fit of the
data over the entire range as demonstrated in Fig. il. Parameter values of n' = 41,
7 = 0.10 and n = 1.5 were irseci.
The method of curve-fitting used is detailed in Appendix G.
If we reverse the process of "Fechnerian integration" (cf. Section 2 3 - 3 2 ) , ie.,
beginning with the psychophysical Law, we differentiate it with respect to intensity
and replace the differential qiiantities with their finite ciifferences, we obtain
Using Fechner's assumption of the constancy of the jnd (A* = Cil, see Eq. (12)),
we can rewrite Eq. (29) as
in agreement with the empirical Weber haction eqiiation obtained by Knudsen and
Riesz (see Eq. (15)).
Figure 11: Same data as Fig. 6 (Stevens, 1969) fitted with the classical entropy Eq.
( 2 6 ) .
Furthemore. the laws of Bloch? Piéron and Blondel-Rey can all be derived from
Eq. (25). albeit with greater effort. It suffices to mention here that the entropy
eqiiation h a been quite successfd at umfying many empirical results.
3.1.2 Failures of the Classical Theory
However. the classical entropy theory also had its few shortcomings. We now docii-
ment some of its shortcomings and failures.
The classical theory only works with stimuli of constant intensity. It cannot predict
the sensory response given a tirne-vmying input.
When we examine the limit of long stimulus duration in Eq. (25), we see that
F adapts to completion (F = O). However, a quick glance at Fig. 2, or at results
obtained from any other sensory modality (e.g. Matthews, Ml), fails to co&m this
prediction.
Fiirthermore, in Fig. 2, we see neural activity even in the absence of external input.
However, if we set I = O in Eq. (25), we find that F = 0.
The classical entropy equation is incapable of predicting the rising portion of the
adaptation c ime , or the effect of de-adaptation as observed in Fig. 2.
Recall that, experimentally, it has been observed that the differential threshold or
Weber fraction decreases for increasing stimulus durations (see Section 2.3 -3.2). In
Eq. (29), recall that = p'lt. Inserting this expression into Eq. (29) shows that the
di fferent i al t hres hold in creases for increasing stimulus durat ion.
In addition. the classical entropy theory fails to predict (or fails to predict rvell)
many of the empirical results of threshoid and simple reaction time phenornena.
At this point. it was evident that we required a more general theory rvhich coidd
address many of the shortcornings of the classical theory.
3.2 Universal Entropy Theory (1993-)
In this section, we s h d derive the central result of the Universal Theory - the gener-
alized entropy eqtiation. This eqiiation is capable of predicting the neural response,
F. given any time-varying input 1. With a few modifications, we are following the
derivation of the original (classical) entropy equation as put forth by Norwich in
1977. Most of the residts of this section have been published in a recent publication
(Norwich and Wong, 1995).
3.2.1 Deriving the Universal Entropy Equation
Fliicttiations may reflect themselves as a stationary time series. Fluctuations generate
imcertainty in the mean of the series.
At the neural level, microscopic fluctuations in stimulus intensity (eg . density
fltictuations) generate uncertainv in the mean level of stimulation. Since the organism
is interested in the mean intensity (cf. Schrodinger, 1989), we might attribute a degree
of "imcertainty" or entropy to the sensory receptors. The receptors are regarded
as reporting their state of uncertainty to the brain. It is hypothesized that the
imcertainty is encoded by the firing rate of the sensory neiirons.
We regard the sensory receptors as sampling the stimulus environment. Successive
sampling of the stimulus produces a time-series of intensity values. For steady input,
a stationary tirne-series of sample values with stimulus variance, a*. and mean, p,
is obtained. This variance then characterizes the receptor's uncertainty in the mean
st imidus intensity. .4ny expression of the entropy shoidd be a monotonically increasing
hmction of the variance. That is, greater stimidus variance gives rise to greater
imcert ninty or entropy.
We mode1 the uncert ainty mathematically using the Boltzmann-Shannon measure
of entropy (Shannon, 1948),
where H is the imcertainty or entropy and p(x) dx is the probability density for the
receptor to sample an intensity value between x and x + dx. Following Brïllouin
(1962), information is regarded as negentropy. That is, reduction in uncertainty will
generate information.
The use of the term "entropy" might suggest to the reader that H is a measure of
physical o r thennodynamic entropy. PVhile Eq. (31) certainly resembles Boltzmann's
H-fimction, no direct relationship between informational and physical entropy was
implied by Shannon. Similarly, our use of the term "entropy" within the sensory
42
domain does not necessarily imply a connection to entropy as iised in physics. although
in certain. restricted cases, sensorid and thermodynamical entropy have b e n s h o w
to be eqiiivalent (see Wong, 1993).
Defining sorne iiseful terminology, we take each sample to consist of many separate
sanzplings. Together, m sarnplings create a sample of size m. The mean of each sample
will tend to be normally distnbilted in accordance with the central Mt theorem.
This property is. of course, independent of the original stimulus distribution (be it
imiform, Poisson, etc.). If 5 is a random variable representing the mean of a sample
of size m, then its density function pif) dZ is given by a normal distribution with
mean ps and variance G/m:
where m is the sample size.
Using Eq. (31), we can now obtain a simple expression for the entropy of mean
stimuliis values. Substituting for p(x) hom Eq. (32) we obtain
For increasing sample sizes, entropy decreases rnonotonically suggesting that the r+
ceptors are losing imcertainty in the mean intensity.
Transmission of sensory information is always iimited by the presence of other
signals or noise. We assiune that these signals have a normal or Gaussian spectrum
43
of intensities and are termed the reference input. Since the receptor cannot distinguish
between the stimulus and the reference signal, the sensory input, as sampled by the
receptor, is actually a convolution of two signals. Denoting the mean and variance
of the reference distribution by and a i respectively, the convolution of the two
normal distributions is another normal distribution (eg. Fraser, 1976) with mean
ps + and variance o$/m + ai. Hence, the entropy of both signai and reference is
Siibtracting off the entropy of reference input alone, 4 In ( 2 r e a i ) , we obtain
This eqiiation then gives the information which is transmitted to the brain by the
sensory receptor regarding its rmcertainty in the mean intensity when rn sarnplings
of the stimuliis have been made.
As i t stands, Eq. (35) is a complete equation which goverm the neural response.
However, three additional postdates are required before this equation can be used to
analyze e.xperimenta1 results.
In most physical systems, the variance is related to the mean. For example, statis-
tical mechanics predicts that the fluctuation in the density of a dilute, classical gas
has variance equal to mean density. Here, we assume that the variance can be related
44
to the mean by the relationship
where E is a constant of proportionajity and p is a parameter of the physical systern
characterizing the magnitude of fluctuations at the receptor level. p is a positive
constant. Eq. (36) states that larger fluctuations are associated with larger quantities.
The term, 61: may be any natiually occurring internal signal within the sensory
system (eg. spontaneous otoacoiistical emissions in the ear. thermal noise, etc.). As
a conseqiience of 61, fluctuations are recorded by the receptor even in absence of an
external signal.
61 is an important parameter within the universal theory. More will be disciissed
in the later chapters. However, nie do rnake the passing remark that the magnitude
of 61 is greater than the absolute threshold, AI. No theoretical reason is given for
this remark except that this is what is usudy observed when the equations are fitted
to data.
Taking f l = cri , a constant, we can now write
R e c d that rn is the sample size. In order that the receptor can process information
from
m is
the stimuliis samples, i t must have the ability to store the samples locally. Hence,
also a measure of local receptor memory. At stimulus omet, we assume that
45
the receptor progressively increases the sample size over tirne. Hence m increases.
thereby reducing the stimulus uncertainty as quantifieci by H. If m were to increase
indefinitel- the receptor would require infinite memory. Hence, it is reasonable to
assiune that m wiL grow to a saturation level, Say m.,, after which the sample size
will no longer increase.
If theory is to match the experimental results, we have foimd i t necessaq to take
rn,, as a rnonotonic increasing hinction of intensity in the form
where O < q c 1 and mzq is a constant to give the correct units for m.,.
Thus. m,, zs the requzred mernosy to store sarnples ut zntenszty I .
In audition and vision, where I may change by several orders of magnitude (eg.
IO"+ for audition)? q is necessary to keep the memory change within reasonable
limits. Notice that me, is dependent on both I and 61.
LWde Eq. (38) appears to be an arbitrary assimption, in Appendk A I present a
physical mode1 which may account for this power dependence.
When the input intensity changes, so too will me,. How does rn approach the new
me, value? Since it is assurneci that memory changes continuousIy, rn will approach
me, smoothly. Let us Say that the rate of change of m is a fimction, f , of the ciifference
between m and me,,
d m -- d t - f (m - m.,)
Expanding f in a Taylor series and neglecting terms beyond the first order, ive obtain
where f (O) = O to allow for steady state when m = m.,. We set a = -/'(O), where
the parameter a. which has the dimensions of inverse time, is expected to be dways
greater than zero.
LWle this derivation is suitable only for s m d merences, our experience with
fitting esperimental data has shown that Eq. (40) tends to work quite well under d
conditions.
A physical mode1 of Eq. (40) is given in Norwich and Wong (1995, Assembly-line
blodel). A possible electrochemical explanation of Eq. (40) might also be found in
the work of Schroeder and Hall (1974).
Eq. (40) can be solved easily if ne, is independent of time. However, if I is a
fimction of time (say, a stimulus with a slowly varying intensity profile), me, will
indirectly be a function of time as well. Under these circtimstances, the solution
becomes
m(t) = rn(t') e-a(t-t') + a e-=' 1 earmeq d i , (41) t'
where the initial condition is evaluated at time t = t'. In principle, we are able to
accoimt for any time-varying sensory input.
Al1 that remains is to relate stimulus imcertainw, H , to sensory response. In
47
1977. Norwich postdated that the firing rate, F' of the receptor's associateci prirnary
afferent neuron is directly proportional to its uncertainty (Norwich. 1977). That is,
wtiere k is a constant and has iinits of inverse time or frequency.
One must pause to consider the importance of this single equation. Whereas the
Boltzmann-Shannon measure of entropy may be considered a modei of the uncer-
tainty resident at the receptor, Eq. (42) is a fundamental conjecture regarding the
objectivity and measiirability of sensation. The tme test of t hese eqiiations, of course,
is in their predictive power both in theory and in experiment.
.4s we s h d see, Eq. (42), dong with Eq. (37), provides a good theoretical pre-
diction of the neural sensory response to all inputs for intensities below physiological
saturation levels.
3.2.2 Summary
Only three eqiiations fÎom this chapter will be required for the remainder of the thesis:
Eq. (3'7), relating receptor uncertainty to stimulus intensity and memory,
Eq. (41) , solving for the receptor memory,
and Eq. (42). relating receptor imcertainty to neural response:
F = kN.
Chapter 4
Neural Explorations with the Uni-
versa1 Entropy Equat ion
If the universal entropy equation zs to supersede the restricted or classical
equation as the equation of p~eference, it must be capable of euen greater
predzctiue power. In the folloving sections, we shall utzlize Eqs. (37) and
(42) to ezplore neural phenornena in a rnanner not previovsly possible with
the simpler classical entropy equation. That zs, we shall attempt to address
the shortcomings of the classical theory mentioned in Section 3.1.2.
4.1 Neural Adaptation
Consider the neural response to a step input as shown in Fig. 12. The stimulus is of
constant intensity, 1, beginning at time t = O. The solution to Eq. (40) is
If the receptor was completely adapted prior to the input then m(0) = me,(,=, =
m& 619 from Eq. (38), and m,,l = mzq (1 + 61),. Obtaining F fiom Eqs. (43),
(37) and (42). we obtain
where 3 = , ' lm&. It is easily seen that mzq always factors out of m and can be
incorporated into ,O.
The denominator in Eq. (44) can be rearranged to give ( I +61)q - rëaC7where
ï = (1 + 6 I)q - 61s. Since r > 0, the denominator is monotonically increasing and,
as a conseqiience, F decreases monotonically for increasing t corresponding to neural
adaptation. Whereas, in Eq. (25), F always adapts to completion (F = 0) for large
t , Eq. (44) shows that, for t - co,
where n = p - q. That is, F does not M y adapt.
4.2 Spontaneous Neural Activity
if there is no external signal (1 = 0) and the receptor is completely adapted, we have
frorn Eq. (45)
Figure 12: Step input intensity.
The presence of 6 1 prodiices a "spontaneous" firing rate independent of the extemal
inptit . as commonly O bserved experiment d y .
4.3 Early Rise in the Adaptation Curve
There are several ways to interpret the early rise in the neiiral adaptation curve (cf.
Section 2.2.1.1).
4.3.1 Correlated Receptor Sarnples
The rate of change of m with respect to time indicates how quickly the receptor
samples. In terms of adaptation, we see that dn t ld t is greatest at initial stimulus
onset. decreasing steadily for increasing durations. It is assumed that each receptor
sampie is statistically independent of the previous sample. However, if successive
sarnples are separated by a period les than the correlation time of the stimulus, the
samples will be not be entirely independent.
The sample variance obtained from a correlated sequence of stimulus values will
tend to be smaller than the variance obtained kom independent samples. Thus, by
Eq. (33), the entropy or response will be diminished as a consequence of the low
variance. .4s the sample values become less correlated, both the variance and the
response will rise. When the sampling period exceeds the correlation period, the
entropy will fall monotonically as expected with adaptation.
The effects of correlation in the olfactory system were explored in a paper by
Aebersold, Norwich and Wong (1993). In this paper, the correlation period was
stiidied tising computer simulations of the Brownian motion of odorant molecules.
4.3.2 Effects of Stimulus Rise Time on Adaptation
Eq. (44) was obtained for a step hinction input at time t = O. More realisticdy.
stimuli have finite rise times which may affect the neural response. For example,
Smith and Brachman (1980) found that by decreasing rise tirne, the peak of the
adaptation curve rises, and vice versa. Using a numerical solution of the differential
eqoation, it can be shown that the iuiiversal theory is capable of predicting this effect.
Suppose that we can mode1 the rising stimulus profile with a function of the form
wliere a is the inverse rise time constant. We have assumed that the stimulus begins
at time t = O. Eq. (40) can be solved numericdy with I ( t ) for different valiies of
a. Siibstituting these values into Eq. (37), we can now observe the effects of the rise
time on the adaptation cime. Using parameters values k = 2, = 0.1, 61 = 0.1,
p = 1, q = 0.5, a = 5 and Ï = 5, the results are shown in Fig. 13.
It is essential to remember that the parameter a (stimulus rise time constant) is
not related to the parameter a (adaptation time constant).
The cime rises to a peak before falling, in agreement with Fig. 2. Furthemore,
we see the peak shift upwards and towards the origin as the rise time is reduced in
agreement wi t h experiment al O bsemtion.
4.4 Neural De-adaptation
If the neural response is adapted to a stimulus of intensity I by time t = 0, the
response takes on the form of Eq. (45). If the signal is now removed ( I = O ) , we can
solve for the shape of the neural response.
R e c d that Eq. (43) is the solution of the differential Eq. (40) for constant
intensity. From Eq. (38), we see that m(0) = m q (1 + 61)' and m, = m:, b1q.
Siibstitiiting these values into Eq. (43), we obtain
Generally speaking, the stimulus intensity I is always much greater than the in-
ternal signal 61. Therefore, m(0) is much geater than me,. Since m(t ) changes
continiiously from m(0) to me,, we see that m(t) decreases monotonically from t = O
onwards.
From Eq. (&), F drops suddenly at t = O because I = O . Afterwards, F increases
steadily as m(t ) decreases with t h e . F keeps increasing until steady state is reached
in Eq. (48). In that case, F approaches the spontaneous value given in Eq. (46).
This residt is in agreement with the experimentdy observed de-adaptation effect
(see Fig. 2).
Figure 13: The effects of stimulus rise t h e on adaptation as predicted by the universal
equation. a is the inverse rise time constant. For top to bottom, ru = 1000,50, and
10 respectively. A shorter rise time produces a higher and quicker peak.
4.5 On the Relationship Between the Classical
and the Universal Ent ropy Equat ion
As mentioned earlier, the heritage of the more complex universal entropy equation
can be traced directly back to the classical entropy Eq. (25). Hcjw can we complete
t his correspondence?
Since the classical equation is lirnited to step inputs as shown in Fig. 12 (see
Section 3.1.2) , we return to Eq. (44), which is the neural response to a step input
as predicted by the iuiiversal equation.
CVe can simplify Eq. (44) for the limit where the stimulus is of shorter duration
( t « l / a ) and higher intensity (1 » 61) to obtain, approximately,
Setting K = k / 2 , = 4 / a and n = p - q, we see that the classical entropy Eq. (25)
is recovered.
4.6 Further Experimental Validation of the
Universal Entropy Equation
In this section, we utilize Eq. (44) to analyze data from real experiments. However,
before proceeding, one shoidd take cognizance of the difficulties in validating such a
complicated equation as Eq. (44). The problem involves the number of parameters.
Wihile only two parameters are required to uniquely spec* a straight line, adaptation
58
riwes woidd typically require 2-3 parameters. Fitting a cime with 6 parameters
throiigh adaptation data would yield no imique set of parameters.
The or&- way of obtaining robust parameters is by sznzultaneously fitting the epua-
tion to multiple sets of data. For example, Eq. (44) is a huiction of taro variables:
that is, F = F ( I . t) . If we have data fkorn the same preparation which measures both
F as a fimction of 1 for constant t and F as a function of t for constant 1, then we
can evaliiate the 6 parameters for two cuves , or 3 parameters or degrees of fkeedorn
per cime.
Cime-fitting of a single. multi-parameter fimction to multiple sets of data using
the same parameter values is a very stringent test of the validity of an equation.
Sirniiltaneoiis cuve fitting has, to our knowledge, never been iised by other groups
to test any sensory theory. For further details, please see Appendix G.
4.6.1 Auditory Neural Response
Brachman and Smith (Figs. 4 and 6, Smith, 1988) measured both firing rate as a
fimction of tone duration at k e d intensity (adaptation) and firing rate as a hinction of
tone intensity at ûxed durations (driven response) . Both experiments were conducted
on the same auditory nerve fibre in the same organism.
In the adaptation experiment, they recorded the number of measured impulses
in a 960 ms bin as a function of stimulus duration in the anesthetized Mongolian
gerbil. The impulse count was converted to firing rate by dividing through by the
bin interval. An averaged firing rate over 91 triais was then obtained. Fig. 14 shows
their transformed data for a 39 dB SPL tone at the characteristic frequency (CF) of
the fiber (2.44 kHz).
For the driven neural response, they measured the maximal hing rate during a one
miilisecond interval after response onset as a function of stimulus intensity. Since the
derivation of Eq. (44) demands that the duration of the stimulus be kept constant,
Eq. (44) is not completely suited to their experimentd procedure. Nevertheless. we
shall use it as a good approximation to their measurements. The data £rom the graph
by Smith (Fig. 6, 1988) have been digitized and are shown in Fig. 15.
Since the same fibre was iised in both experiments, we require that the same set of
parameter values should fit both sets of data. Hence, Eq. (44) has 6 parameters (k,
, . p. , a, 1 ) For the driven response, we require an additional parameter f whch is
the average stimulus diration. The values k = 3.3 x 102 spikes/s, ,O = 2.5 x 10-~ IL',
p = 1.1. q = 9.0 x IO-', a = 9.0 x 10 Hz, 61 = 2.5 and t = 1.1 x 10-l rns were obtained
by simultaneously fitting the data with Eq. (44). The theoretical fit is shown as the
solid line in Figs. 14 and 15. The fit is quite good in both cases. In Fig. 15, we see
that, for the higher intensities, Eq. (44) fails to predict the saturation of the neural
response.
We can also predict the cat muscle fibre response to an increasing ramp and hold
Figure 14: Neural adaptation data of Smith (1988) fitted with Eq. (44). The same
set of parameters were used to fit the data in Fig. 15 as weil.
Figure 15: Driven neural response as measured by Smith (1988) and fitted by Eq.
(44). Data was coliected fIom the same fibre as Fig. 14. Consequently, the same set
of parameters were used to fit both sets of data.
stimidiis tiiçed by Awisziis and Schafer (NB), the mechanoreceptor response to steady
and sinusoidal stimulation as measured by Bohnenberger (198 1) (see Norwich and
Wong, l995), and also many other input profiles for wious sensory modalities as
well.
4.7 Ot her Investigators
The only other investigator to our knowledge who has worked with comparable gener-
ality on developing a universal theory for sensory receptor action is Zwislocki (1973).
In this paper: he examines the effect of stimulus intensity on the neural response.
His approach is based on detailed receptor mechanisms and is somewhat empirical.
Furthemore, Zwislocki does not deal with adaptation phenomena. However, he does
demonstrate similarities in the neural response for a wide range of sensory modalities
and organisms.
Chapter 5
Unieing Threshold P henomena
In the prevzous chapter, we explored a wide range of neural phenornena using
the universal entropy equation. We now wish to use the same eqvatzon to
study psychophysical thresholds.
However, before tahng to this task, it zs important to survey what m-
searchers currently understand or do not understand abovt threshold phe-
nomena. Quoted belout 2s a passage from a review article unitten with re-
gard to the current state of auditory threshold research (Algom and Bab to f i
1984):
"Ah hough there is general agreement with regard to the fundamental
empirical findings involvecl in the time - intensity relation, there is no such
consensus with regard to a theoretical formulation. Various theoretical in-
terpretations have appeared in the literature over the 1 s t 40 years. These
theoretical formulations have been mainly independent developments by dif-
ferent authors rather than representative of an organic development of a
dpamic field of interest. The interpretations that have been offered differ
in terms of the coverage of data as well as in their level of mathematicai so-
phistication. This situation rnakes it difficult to assess the "state of the art"
reliably: as well as to incorporate new experimental finds into an accepted
body of relevant knowledge."
The sarne evaluation may also be ascribed to the understanding of thresh-
olds for other sensory modalities as well.
In this chapter I hope to rectzh this situation. Bvilding upon the earlier
*work of Norwich (1987, 1989), 1 wish to demonstrate that we now have a
theory capable of unzhing the study of thresholds almost in i ts entirety. mile
the success of the clessical entropy theory in this area has been lzmited, we
shall see that many threshold phenornena can be understood using only Eq.
(37) coupled tvith a single threshold h ypothesis.
While o u seensory receptors are sensitive to single atomic events, the combineci
action of a number of these events is required before a psychophysical response can
be generat ed. Yet , how is the detection of t hreshold stimuli achieved given the neural
response? That is, what condition does the brain impose upon the neural response
before the st imuliis can be detected psychophysicdy?
We shall attempt to answer these questions in the following sections.
5.1 Thresholds and the Neural Response
In Section 2.3.3, the sensory threshold was d e h e d as the minhum stimulus level
which is detectable, Say, 50% of the tirne. Thus, a subject may be reqiked in an
experiment to detect a weak flash of light in a dark room. The experimenter adjusts
the intensity of the flash until it is detectable, Say, half of the t h e . We termed
this the absolirte threshold. For stimulus intensities above the absolute threshold,
the differential threshold governs how closely we can merentiate between stimuli of
two different intensities. As with the absolute threshold, the differential threshold is
always d e h e d with a statistical criterion in mind.
Both the absolute and the differential thresholds exhibit a reciprocal relationship
between minimum perceptible intensity and stimulus duration. Thus, the t h e e main
variables governing t hreshold phenomena are A I (threshold intensity) : At (stimulus
duration) and F (sensory response).
Recd that the organism h s t cornes into contact with the stimuliis at the neural
periphery (cf. Fig. 1). Hence, the threshold condition must ultimately be decided
from the neural response. Eq. (37), introduced in the Chapter 3, relates the neural
response F to both the stimulus intensity I and stimulus duration t: F = F ( I , t).
That is, the three threshold variables are all related in a single equation for the neural
response.
In the following section, we shall examine how the threshold condition can be
denwd from the neural response iising Eq. (37).
I t is essential to keep in mind tliat only Eq. (37) will be required for this entire
chap t er .
Some investigators incorporate signal detection theory in their study of sensory
thresholds (eg. Durlach and Braida, 1969; Hellman and Hellman, 1990). Signal
detection theory is a mathematical and theoretical system that deals with both de-
cisional and sensory components in detection and discrimination tasks (Coren and
PVard, 1989). While this approach is hdamentaily different fkom the entropic a p
proach, the two methods are actudy compatible (see Hellman and Heilman, 1990,
and Wong and Norwich, 1995).
5.2 Introducing the Threshold Hypothesis
Fig. 8 shows one method of measuring the differential threshold. Given a pedestal
of intensity I , what is the minimal increment Al which is detectable by the subject
Say 50% of the tirne? This method of measuring the Weber fraction is cded the
continiious method. Fig. 8 has been replotted in Fig. 16.
In the last chapter, we examuieci the phenomenon of neural adaptation by solving
for the neural response given a step input. (If required, a quick reuiew of Section 4.1
would be helpful at thù point.)
Similarly, we can also solve for the neural response H = F/k correspondhg to the
input shom in Fig. 16. Solving for the response gives the residts shown in Fig. 17.
The detailed calculations are left to Appendix B.
Fig. 17 is in good agreement with the neural response as observed experirnentalb
Please see Smith and Zwislocki (1975).
As mentioned in the introduction, one feature comrnon to aU sensory organs is
their ability to transmit sensory information. Thus, following Norwich et al. (l989),
we can now e.xpress the fhdamental entropic conjecture regarding the detection of
thresholds: An increment AI c m be detected ij and only zf the change in neural
cntropy o r information ezceeds the constant AH. In other words, a fured quantiîy of
information must be transmitted neurdy in the interval [O, At] before the increment
can be detected.
Mathematically, we might say that A I is detectable if and only if
where t , is the time of first receptor sample in the interval [O, At]. H is given by
where ,O = B'/m&, Y = AI/ ( I + 61) and C = B ( I + 61)". Eq. (51) is derived in
Appendk B and corresponds to the neural response to the increment in Fig. 16.
The threshold hypothesis is sornewhat analogous to the classical Bohr approach to
the atom. Given an electron in its gound state, what are the energy requirements to
68
Figure 16: S tirnulus input for continuous increment .
Region 1
Figure 17: The neural response corresponding to Fig. 16.
move the electron to the next orbital or higher energy state? If a photon is incident
iipon an atom with less energy than the difference between the two energy states,
the electron will stay in its current orbital. Only a photon with su8icient energy can
move the electron to the next orbital. Sirnilarly, our hypothesis states that the brain
does not respond to any stimulus which fails to produce a change in neural entropy
exceeding AH.
CVe now explore how this single hypothesis can change the entire way in which
thresholds are interpreted.
5.3 Differential Thresholds and Weber Fractions
5.3.1 Deriving the Different i d Threshold Equat ion
Specid cases of Eq. (50) may be expresseci in simpler forms. We make the following
asstunptions in order to simpli@ the equation: A I << I + b l , At >> t,, and A H « 1.
R e c d that AI is the differential threshold (see Section 2.3.3.2). The assiunption
AI << I + 61 implies that the merential threshold intensity is s m d compared to
the combined input (external + intemal signal). The second assumption relates
sampling time to increment duration. Qualitatively, these two assumptions appear to
be compatible. Recall that there is an inverse relationship between stimulus intensity
and stimulus duration for a threshold response. A longer duration would imply a
smaller incrernent. and thus the two assumptions are indeed compatible.
The third assimption relates to the absolute magnitude of the transmitted infor-
mation. Since threshold quantities are relatively srnail, it is reasonable to assume
that the magnitude of AH, or the threshold criterion, will also be srnall.
It is important to point out that AH is directly related to the statistical criterion
established by the experimenter (see Section 2.3.3.1). A H links the entropic approach
to the theory of signal detectability. While there exists no current theory which
can relate the two quantities mathematically, we know that A H is a monotonicdy
increasing hmction of the response criterion. That is, less transmitted information is
reqiiired if the experimenter sets a lower response criterion.
Taking a series expansion to first order in both AI/ (1 + 61) and AH in Eq. (50),
one obtains
where we have set n = p - q. The details of this calculation are preçented in Appendix
5.3.2 Implications of the Differential Threshold Equation
Recall that there are many empirical observations and equations governing the Weber
fraction (see Section 2.3.3.2). Using Eq. (52), they can be explained quite simply.
This eqiiation is identical to the empirical Knudsen-Riesz Weber fÎaction Eq. (15)
and is simiiar to the expressions proposed by many other investigators (ie. Hecht.
1935: Siebert. 1968).
As the pedestal intensity is made smaller and s m d e r (I -+ O), we can expand Eq.
(52) to obtain
In this case. a double logarithmic plot of A I I I vs. I will yieid a straight line Mth
slope - 1 as observed by many investigators.
5.3.3 Validating the Different i d Threshold Equation
Here. we are once again dealing with an eqiiation of many parameters. In total, Eq.
(52) contains 5 parameters ( A H / q is considered as a single parameter). We shall use
the technique of simultaneous curve fitt ing to validate this eqiiation.
In Eq. (52), the Weber haction is a hinction of both I and t. By curve fitting
simidtaneoiisly to both variables, we are subjectzng this equation to a uery severe
experimental test.
5 -3.3.1 Visual Differentid Thresholds
Roufs (1971) rneasured the differential threshold of the eye to point sources of light (1°
73
fovea). Both the pedestal intensity (1) and the increment duration (At) were varied.
In Fig. 18' Roufs' measixements (symbols) along with the theoret ical prediction
(solid line) are plotted for A I vs. At. I is held constant as a parameter and its values
are listed in the Iegend.
The pedestal intensity I is measured in units of troland (td). The troland is
dehed as the retinal illumination from a 1 candela/m2 light source when the area of
the aperture of the eye is 1 mm2.
Jiist as a reminder. it is important not to confise the differential threshold? AI ,
with the Weber fraction, AI/[ . In Fig. 18, AI is plotted.
Except for a, notice that the set of 9 curves can fitted with a single set of parameters
( A H / q = 1.0 x IO-*, n = 7.5 x IO-'! 61 = 1.8J = 1.1).
The reason why different a values were used to fit the data can be understood quite
simply. The parameter a determines how quickly adaptation proceeds. In the visiial
system, there are two types of receptors. The rods and cones each have a different
time coiuse for adaptation. yet both function simiiltaneously in the detection of the
thresliold stimulus. Hence more than one value of a is required for the hybrid visual
receptor described by Eq. (52). The value of a = 4.4 x 10 Hz was used for the top 3
ciwes, a = 2.5 x 10 Hz for the middle c w e , and a = 1.6 x 10 Hz for the remahing
cimes. The valiies of a correspond roughly to the different visual conditions: photopic
(vision iuider daylight or bright illumination, cone vision), mesopic (vision under
Figtire 18: The visual dEerentia.1 threshold data of Roufs (1971). Theoretical fit iising
Eq. (32). Notice that 9 cunres were fitted to a total of 7 parameters. The unit 'td'
stands for troland.
rediiced illiunination. combined rod and cone vision). and scotopic (vision under low
illiiminance. rod vision).
In total. 7 parameters were used to fit 9 ciwes. This corresponds to l e s than one
degree of freedom per curve.
5.3.3.2 Auditory Differential Thresholds
Eq. (52) can also simultaneously fit data obtazned /rom two dzfferent experiments.
In the first e-xperiment (Carlyon and Moore, 1986, 'CM'), the Weber haction was
measilrd as a fimction of I for a 500 Hz tone (data from Fig. 1 b, continuous pedestal
condition in the absence of bandstop noise). The duration of the increment was 20
ms. That is. AI / I was measiired as a hinction of I for At held coustant.
The second experiment involved measuring the Weber fraction as a fimction of
increment duration (Garner and Miller, 1947, 'GM'). Using a 500 Hz tone, they made
measurernents at both 40 and 70 dB SL (Table 1, data of GAII). That is, AI/I was
measured as a function of At for I held constant.
CVe simdtaneoiisly fitted Eq. (52) with 6 parameters to three sets of data, or an
average of 2 parameters per data set. 61 was h e d at 2-5.?threshi where Ithresh i~ the
absolute threshold measured at 500 Hz (see Appendix E). We leave the discussion of
this point to Chapter 8.
Ithresh is temporarily used to denote the absoliite threshold (see Section 2.3.3.2) to
avoid confùsion with the ditferential threshold A I (see Section 2.3.3.1).
3. n. a. are the first three parameters; A H l / q and A H z / q are the 4th and 5th.
Since the thresholds were determined by two different criteria (CM. 71%; Gkl, 30%),
tsvo different values of A H were required. We set iip the equations as follows:
' A H 2 q ) 1 + 2-51 1 ( 1 0 5 ~ ~ ~ ) (' + ,3(105-46710+?.5&hreSh ) (GM, 40 dB SL) y
1 (l + w) (l + ) (GM, 70 dB SL)
( 5 5 )
where we have converted al1 intensity values to SPL iising standard threshold data
(see Appendiu E).
The resiilts are shown in Figs. 19 and 20. The following parameter values were
obtained: A H l / q = 1.6 x 10-~, A H 2 / q = 8.1 x 10-~, ,8 = 7 . 0 ~ I O - ~ I ; ~ , n = 3.6 x 10-1
and a = 1.1 x 10 Hz. Five parameters were used to fit 3 curves.
From the parameters, we see that AHl > A H 2 , in qualitative agreement with the
higher criterion iised by Carlyon and Moore.
In Fig. 20, the c w e s deviate from the data for s m d At, which may be seen in part
as a violation of the inequality A I « I+61. However, even when this approximation
is not made, the theory still deviates fkom the data. Since the hequency spectrum
for short tones is broader by the uncertainty relationship (A f At = 1/2), it has
been stiggested that the siibjects in the G m e r and PvIiller experiment were using, in
part? the high frequency spectral cues. The extra stimulus artifact may have given
the siibjects a substantial advantage in detecting the shorter increments - hence the
77
Figue 19: Data of Carlyon and Moore (1986) as fitted by Eq. (55) showing Weber
fraction as a fiuiction of intensity. The same parameters were used to fit the data in
Fig. 20.
Figure 20: Data of Garner and Miller (1947) as fitted by Eq. (55) showing Weber
fraction as a fimction increment duration. The same set of parameters were used to
fit the data in Fig. 19.
spiirioiisly low values for the Weber fractions.
5.3.4 A Generalization of the Threshold Hypot hesis
We now retiirn briefly to the experiments of Roufs. An obvious generalization of the
e.uperimenta1 paradigm as shown in Fig. 16 is to rneasure the threshold associated
with a decrement. That is. instead of a change of +AI, we ask the subject to detect
a change in the pedestal intensity of -41.
What might we expect £rom the residts of such an euperiment? If we examine Eq.
(52), we see that a first order expansion in both AI/ (1 + 61) and A H produced an
eqiiation which is linear in both A l and AH.
Thiis, if we postdate that a decrement of - A i c m be detected if and only if
the accompanying n e i d response shows an increase in uncertainty or entropy of the
amount - A H , then Eq. (52) predzcts that such an ezperiment would yield identical
results to the corresponding incremental expenrnent (apart from the sign o f AI).
That is, a loss in information is required for the perception of a decrement.
Not siirprisingly, this is what was observed experimentally. In Fig. 21, Roub'
residts are plotted showing bath the data for the incremental and the decrernental
experiments for the same pedestal intensity. The agreement is quite striking.
5.4 Absolute Thresholds
The absoliite threshold is the limit of the differentid threshold when the pedestal is
rediiced to zero (1 = O) (see Section 2.3.3.2). Frorn Fig. 16, all that remains of the
intensity profile is a stimulus of magnitude A I and duration At.
Reti~rning to Eq. (52), midtiplying both sides by I and setting I = O, we obtain
where
Eq. (56) was Brst obtained by Plomp and Bouman (1959)' and later by Zwislocki
(1960) as an idternative expression for the Blondel-Rey law (or Hughes' law in aiidi-
t ion).
Of the pantheon of empirical relationships governing threshold phenornena, Eq.
(56) appears to be the only equation which has ever been derived. Even in this
case, the derivation is based iipon empirical equations governing the neural response
(Zwislocki) or some arbitrary mode1 of the sensory organ involving electric circuits
(Plomp and Boiunan).
As long as the same response criterion is used, a single value of A H determines
81
Figure 21: Data of Roufs (1971) demonstrating that the magnitude of the Merential
threshold is the same for both a positive and a negative increment for the same
pedestal intensity. The open symbols are for positive incrernents, the med syrnbols
are for negat ive increments .
both the differential and the absolute threshold. Returning to Fig. 18: we see that
the single Eq. (52) fits ail c w e s includiog the c u v e for I = O using only a single
value of AH. The cime for 1 = O measiires the absolute threshold.
5.4.1 Bloch's Law: Part 1
When At « l l a , we c m expand Eq. (56) to obtain, approximately,
which is Bloch's law. although care must taken to observe that the approximation
AI << 6 1 remains valid. We s h d return to Bloch's law fkom a Werent perspective
in the next chapter.
Chapter
Simple Reaction Time
Recall that reaction time zs another class of phenomena which exhzbits an
inverse relationshzp between stimulus intensity and stimulus duration. It zs
not usually classijied arnong threshold phen cmena. Nevertheless, we endeau-
our to show in this chapter that there zs a threshold assoc2ated with simple
reaction time, and that absolute, differential and reaction thresholds are al1
related.
6.1 Deriving Piéron's Law
The measurement of simple reaction times typically follows a protocol where a subject
makes an motor response (say, by pressing a button) immediately upon detecting the
stimulus input shown in Fig. 22. At, is c d e d the simple reaction tirne.
Apart hom the motor response, the measurement of simple reaction times is iden-
tical to the measirement of absolute thresholds. We can follow upon this idea by
Figue 22: Stimulus input for a simple reaction time experiment. Siibject responds
by making a motor response at t = At,.
retiirning to the t hreshold hypothesis introduced in the 1 s t chapter.
CVhile most absolute thresholds intensities are s m d compared to the fidl phys
iological range. intensities associated with reaction time measurernents tend to Lie
within the normal perceptible levels. That is, L I is almost always much greater than
the absolnte threshold intensity.
With this mind, we must now discard one of the assumptions of the last chapter
- that A I is small. Instead. we adopt a new approximation that A I » 61. By the
reciprocity relation. since A I is now large, At shodd be smaiIer in compensation.
We miist be very careful to differentiate properly between At and At,. As postu-
lated by Eq. (19), the reaction time At, is actually a sum of the sensory processing
time At, and the time required to make a motor response At,. WhiIe At may be s m d
in magnitude, At, might be substantial in cornparison. Caution must be observed
so that we do not wrongly estimate the order of magnitude of these variables.
In the 1 s t chapter, only the sensory processing tirne (At ) was required since there
was no rnotor response.
The other approximation, that t , « At, must also be discarded since At is now
much smaller.
Retiiniing to Eqs. (50) and (51), we now set I = O (no pedestal is present). This
time we assume that A I » 6 1 and A H « 1. At is left unrestricted. Solving Eq. (50)
for At and expanding the resulting equation to qth order in 61/AI and h s t order in
AH. we obtain
(59)
where we observe that O < q < 1 and p > q (see Section 3.2.1). The detailed
calculation of Eq. (59) is provided in Appendix D.
Wlile this eqiiation appears daimting, it is a c t u d y qiiite simple. D e m g
and assiiming that 6 I / A I zz O (ie. we take e zeroth order expansion in 6 I / A I of Eq.
(59)), we get
Q At = At,,, + - AIn '
where 9 = 2AH (1 - ëa'0)2 /a,5ëat0.
As mentioned earlier, this equation governs the t h e required for the sensory system
to process a stimulus of intemity AI.
Eq. (61) is fascinating because it represents a limit or Iower bound on the process-
ing time of the hiunan sensory system. That is, no rnatter how large we make AI,
the sensory system caanot process the information any faster than Atmin.
If we were to obtain an expression for the simple reaction time fkom Eq. (61), we
woidd add the rnotor response time, At,, to both sides of this equation to give
e At, =At,,,+- AIn '
where Atr = Atmin + At, (cf. Eq. ( 19)). Eq. (62) is Piéron's law, fkst introdticed
in Section 2.3.4.
It is proposed that g be calleci Piéron's constant in honour of the eminent physiol-
ogist .
6.2 Implications of the Entropic Derivation of
Piéron's Law
R e c d that one of the interesting features of Piéron's law is its simihrity in mathe-
matical form to the Knudsen-Riesz Weber &action Eq. (15). We can now understand
why this is so: both equations are a linearization of Eq. (50) with respect to A H and
I or AI. Consequently, the exponent (n) is identical in both equations (see Section
2.3.4).
Yet, we must remember that Eq. (62) was derived with the rather poor approx-
imation that 6 I / A I = O (zeroth order expansion). This assumption is not always
valid. üsing instead E,q. (59) for At, one obtains
e O 1 At, = At,,,, + - e 2 +-+- Ain AIp AIg'
where and fi are a l l positive constants.
m3en data which conforms closer to Eq. (63) are "forcibly" fitted with Eq. (62)
(Piéron's law) , we see that a much lower exponent would be obtsined kom the fitting
procedure in order to compensate for the two missing positive terms in Eq. (63).
88
This is exactly what has been observed by many researchers incltiding Marks (1974)
and Norwich (1993). The "effective" exponent obtained by fitting Eq. (62) to exper-
imental data is almost always lower than the exponent obtained by fitting Eq. (25)
to Weber fraction data for the same sensory modality.
6.2.1 Bloch's Law: Part II
Only one empirical resiilt from Chapter 2 remains unanswered.
Recall that a variant to Bloch's law was proposed by G m e r in 1947 (cf. Eq. (7)).
This law is approximate at best when used to fit experirnental data. Nevertheless.
we shall attempt to derive this equation although, admittedly, the derivation is not
ent irely satisfact ory.
Rewriting Eq. (55), we obtain
If A I is srnall (although not too small as to violate the assumption 61/Ai << 1), At
is mtich greater than Atmin. This, we can drop this term in favour of the remainùig
terms. Generally speakuig, n, p and q are all of the same order of magnitude. Hence,
we might approximate Eq. (64) with the simpler equation
where 0 is essentiaily the mean of the three exponents. It is important to keep in
mind that Eq. (65) is a very crude approximation to Eq. (64).
89
Rewriting Eq. (65) in more familiar form we obtain
In Chapter 5, Eq. (58) (Bloch's law) was derived under the condition Al << 61.
That is. A I is ciose to threshold Levels. By contrast, Eq. (66) was derived with the
approximation Al >> 61: that is, AI is at suprathreshold values. Hence, we might
siunmarize the resiilts of both chapters with the foilowing table:
A l At = C,, AI near threshold ( A I < 6 1 )
A I atl/' = C31i A I suprathreshold (AI >> 6 1 ) ,
a residt which has been observed by many researchers. Please see Algom & Babkoff
(1984).
Chapter 7
Speculat ing on the Psychophysi-
cal Response
This chapter deals wïth the extension of the universal entropy theory to new
domains. The results of this chapter repi-esent the "state of the art" and,
consequently, require fvrther investigation and research.
A11 of the psychophysical phenomena considered thus far have been in a sense
"objective". That is, the measurement process does not require an arbitrary decision
or judgment on the part of the subject or experimenter (apart hom deciding on the
statistical criterion for a threshold) .
However , ot her areas of psychophysics do involve some element of arbitrariness.
For euample, how do we rate the subjective response of an individual? What rnethod
shoidd be used for s cahg the response?
As we proceed, we s h d bear in mind these concem.
7.1 Psychophysical Adaptation
The difficidty of scaling the subjective response to psychophysical adaptation was
overcome in part by the method of Békésy, which provides an objective rneasiire of
psychophysical adaptation. Before we get to his methodology, let us briefiy digress
to his briliiant scientific life.
Georg von Békésy was a Hungarian scientist who obtained his doctorate degree
in physics from the University of Budapest. Working in the Himgarian Telephone
Research Laboratory. he became interested in how the human e u works. There has
never been scientist who had contributed more to the understanding of hearing than
Békésy. For his contributions, he was awarded the Nobel Prize for Physiology or
Medicine in 19610 joining a large number of physicists who have won this prize.
To measure adaptation in the ear to pure tones, Békésy employed a technique
whereby a siibject woidd match the intensity of an adapting tone in one ear to the
intensity of another tone in the opposite or contralateral ear. The tone in the con-
tralateral ear is of fixed duration (200 ms) and the sound intensity in the adapting
ear is of futed intensity (94 dB). The fiequency of both tones are identical (800 Hz).
Let iis denote the intensity of the matching tone as 1, (in the contralateral ear)
and the intensity of the tone in the adapting ear as I.. That is, I, is k e d a t some
value and I, is adjusted at regdar intervals to match the loudness of I.. Békésy's
measiuements (1929) for a mean of 10 subjects are s h o w in Fig. 23.
Figue 23: Psychophysical adaptation as measured by the technique of loudness
matching. Data of Békésy (1929) as fitted by Eq. (69). dB Adaptation is defined as
10 1% (Im/L).
Békés- in another experiment. was able to show that the effect observed in Fig. 23
is not due to an interaction between the h o ears. That is, the rneasirred phenornenon
issues from a single ear alone. However. this is, admittedly, still a contentious issue.
Please see, for example. Scharf (1983).
To obtain a theoretical equation which may accotmt for the data in Fig. 23, we
briefly retiirn to the two classical entropy Eqs. (24 ) and (25). Eq. (24) govems
the psychophysicai response and Eq. (25) govems the neural response. The simple
relationship between these two equations implies that
where we have ignored for the moment the difference in time scale (cf. Section 2.3.1).
We s h d now attempt to use this same relationship to obtain the psychophysical
response from the neural response for the universal theory as well.
The neural response F ( I ? t ) to a step input is given by Eq. (44 ) . Since @ a F, it
does not matter whether we equate Q or F. Thus, for the condition of equal loudness,
we demand that
where 1, = 94 dB, t, = 200 ms and O < E < 1 is an empirical parameter to account for
the difference in time scales (see Section 2.3.1). Simphfying this equation we obtain
ù'sing the values of p = 4.1 x 10-l. q = 9.8 x IO-* and 5 = 9.0 x IO-^. the theoretical
prediction is shown by the solid cime in Fig. 23. dB adaptation is d e h e d as
1 O I I ) a was set qua1 to the value obtained in Section 5.3.3.2 for the
differential threshold (a = 1.1 x 10 Hz), and the value of 61 was pegged at 2.5 times
the absoliite threshold value for an 800 Hz tone (see Appendk E).
Implicitly, we have made the assumption that the same set of parameters should
satisijr both ears. However, we mtst keep in mind that this may not alaays be the
case.
The agreement between the theoretical curve and the data is qiiite striking. Notice
that the value of < predicts that neuroacoiistic adaptation proceeds approximately
1000 times fmter than psychoacoiistical adaptation.
In essence: we have demonstrateci that psychophysical adaptation can be related to
neiud adaptation throiigh a linear transformation in stimulus duration t. However,
liasserman and Kong (1974) have warned against possible errors in correlating the
t;wo responses. In particular, t hey distinguish between psychophysical time ( "stimulus
diiration" ) iuid neurophysical time ( "tirne after stimulus omet"). This merence is
best imderstood in terms of an experiment involving stimuli of short duration . The
netval response reflects an instantanmus reaction to the sensory stimulus, whereas
the psychophysical response might reflect the totality of the entire stimulus (ie. an
integrated response). In the case of an integrated response, psychophysical adaptation
camot be obtained fiom neural adaptation through a simple isomorphic map.
7.2 Magnitude Estimation
In Section 2.3.2. the concept of magnitude estimation and the law of sensation was
introdiiced. We now apply the universal theory towards the understanding of this
psychophysical effect .
Let i i s begin with the Weber fraction.
7.2.1 The Measurements of Riesz
R. R. Riesz was a physicist working at AT&T Bell Labs in the 1920's and 30's. His
name is nearly eponymous with the classical study of auditory Weber fractions. In
Section 5.3.2, his empirical equation was derived as Eq. (53). Using Riesz's spbo l ç ,
we can rewrite Eq. (15) as
where Ithresh is now ~ s e d to denote the absolute threshold to avoid confusion with
the differential threshold AI. A.rranging the terms we obtain
which we showed earlier to be similar to Eq. (52) :
where we have midtipiied both sides of the equation by I / ( I + 61). Since Riesz's
diiration of increment (At) is on the order of 1 sec, we have, approximately?
iising the value of a = 10.56 Hz obtained from Section 5.3.3.2 for the ditferential
t hres hold. Hence,
Riesz rneasured the Weber fkaction at different frequencies (see Fig. 7). By fitting
al1 his cimes to a single equation, he was able to parameterize S,, S, and n as
If we now make the identification that
we can lise Fechnerian integration (see Section 2.3.3.2) to obtain H = H ( I ) . Fechne
rian integration implies that , to a first order approximation,
Siibstitiiting for AH/AI kom Eq. (74, we obtain
Since H is linearly related to F from Eq. (42) and is linearly related to F from
Eq. (67)' we can write for the psychophysical response
where A is a constant of proportionality. Note the similarity between Eq. (81) and
the classical eqiiation for psychophysical response, Eq. (26).
7.2.2 Magnitude Estimation
We can now use Eq. (81) to analyze magnitude estimation data. 0 and n can be
set. to the values given by Riesz's parameterization. Once again, it is assumeci that
6 1 = 2.5 ILhrerhi where the value of the absolute threshold is given in -4ppendix E.
Thus, the only j?ee parameter left in Eq. (81) 2s A.
Luce and Mo (1965) carried out experiments where subjects were asked to estimate
the loiidness of tones using an assigneci number scale. A 1000 Hz tone was used and
so the values of n and ILhrerh were d evaluated at this fkequency.
We take the subject's mean magnitude estirnate to be numerically equal to the
value of a. Eq. (81) was used to fit their data (Fig. 2, Subject 9) and a value of
A = 2.1 x 102 was obtained. The theoretical fit dong with their data is shown in
Fig. 24. The agreement is quite strikzng considering that the parameters in Eq. (81)
98
were drawn from three sepa.rate ezperiments of three dgeren t paradzgms (absolute and
dzfferential thresholds. and n e a n magnitude estimation).
Continuing this process with other frequencies, we would obtain a set of frequency
dependent magnitude estimation or loudness cimes. In a recent paper (Wong and
Norwich? 1995), it was demonstrateci that the phenornenon known as equal loudness
contours can be derived from siich a process.
However, the data in Fig. 24 are by no means representative of other rneasurements
made for magnitude estimation (eg. compare Hellman and Zwislocki in audition,
1961). In general. Eq. (81) camot be used in conjunction with parameters obtained
from other e-xperiments to give a good theoretical fit to all magnitude estimation
data. Perhaps ths discrepancy may be due to problems of psychophysical scaling.
That is, different psychophysical techniques give rise t O different lo~idness cimes. A
paper is currently in press dealing this issue (Norwich and Wong, 1996).
Figure 24: Estimation of loudness. Data from Luce and Mo (1965). The theoretical
fit is from Eq. (81) with only one fiee parameter (a scaling parameter). ,4U other
parameters were obtained from the measurements of Riesz (1928).
Chapter
Discussion
In this chapter we focus on severnl new ideas which have made their appear-
ance in this thesis.
8.1 The Interna1 Signal, SI
Let LIS examine the key equations which contain the intemal signal term 61.
In Eq. (46) (Section 4.2), bI governs the spontaneous activity of the sensory
neliron. If 6 1 = 0, then there is no spontaneouç firing rate. That is, Fsp = 0.
The Weber fiaction Eq. (52) will not fall with slope -1 if 61 = O. This point has
also been made by hellman and Hellman (1995) specificaiiy for audition. To better
imderstand this point, we set 61 = O in Eq. (52) to obtain
This. in the limit as I -+ O, Eq. (82) becornes
AI 1 - O( -. I In (83)
That is. the Weber fraction fds with dope -n. We r ecd (Section 5.3.2) that with
61 > O? A I / I a l/I (Eq. (54)).
The absoliite threshold vanishes if there is no 61. Provided that n 5 1, setting
61 = O in Eq. (57) gives I, = 0, and consequently A I = O from Eq. ( 5 6 ) .
Piéron's law (62) becornes exact if 61 = O. That is, Eq. (62) eqiials Eq. (63) since
91 and g2 both equai zero when 61 = O. In this case, there would be no theoretical
explmation for why the exponent obtained from the Knudsen-Riesz Weber fraction
eqiiation (Eq. (15)) is ditferent h m the exponent obtained hom Piéron's law.
Generally speaking, 6 1 divides the domain of sensory studies into two regions: (1)
differential and absolute threshold phenornena and (2) simple reaction time phenom-
ena. That is, in Chapter 5 (threshold), we took the approximation A I « 61. In
Chapter 6 (reaction tirne), we used the approximation A I >> 61.
In Section 5.3.3.2 on differential thresholds, it was h s t postulated that 61 is ap-
proximately 2.5 times the magnitude of the absolute threshold (AlAT). Furthemore,
EI = 2.5aIAT was iised when the psychoacoustical adaptation data of Békésy and
the magnitude estimation data of Luce and Mo were fitted (Chapter 7).
The value of 2.5AIAT was hrst suggested in the work of Zwislocki (1965), who
incorporateci a term similar to our 61 in his derivation of a psychoacoustic loudneçs
102
ftmction.
We are now in a position to how see closely 61 -- 2.3AIAT matches the experimental
values. First. it must be pointed out t hat AIeAT is a function of sound frequency. That
is? ttie absolute threshold at 500 Hz wiU be different from the threshold at 1000 Hz.
The effects of frequency on the absolute threshold is shown in Fig. 28 (Appendix E).
The values used to fit the data of Garner and Miller, and Carlyon and Moore
(Section 5.3.3.2) can be siibstituted into
This eqiiation was obtained by inserting Eq. (57) into Eq. (56). Typically speaking,
the duration of tones (At) iised clinically to measure the absolute thresholds are such
that 1 - e-"*l = 1 since a = 1.1 x 10 Hz and At is on the order of 1 sec. Substituting
the remaining values into Eq. (84) (including 61 - 2.5AIAT), we obtain, for a 500
Hz tone, AIAT = 17.8 dB SPL with the 50% criterion. The rneasured value according
to Fig. 28 is approximately 14.7 dB SPL. The two values are within reasonable
agreement. If 61 = 2 N A T was used instead, we would obtain aimost exact results
(15.0 dB).
It can be seen that 61 is an important parameter within the sensory system. Per-
haps fi~tiire experimental work will be able to iden te and elucidate the origins of
this term.
8.2 A Comrnon Set of Acoustical Parameters?
It is not sificient to jiidge a theory purely by how well it fits the available data. There
is an additionai reqiiirement that the fitted parameters must cohere between the dif-
Ferent edxperimental paradigms. Thus, we demand that the parameters obtained hom
threshold measurements must match the parameters obtained fkom simple reaction
time. etc.
We have already demonstrated a concurrence of the parameter 61 across the various
acoiistical experiments. Similady, we can make cornparisons wit h ot her parameters
as weU.
Recall that the value of 0 = 7.0 x ~o-~I;' was obtained from simultaneously
fitting the data of Carlyon and Moore, and Garner and Miller (see Section 5.3.3.2)
- for a 500 Hz tone. Frorn Eq. (78), Riesz's measurements predict that ,DI:,,, -
3.6 x 10-~. Using the threshold data from Appendix E, we can solve for fi to obtaui
1.2 x IO-* I;l at 500 Hz. The two values are of same order of magnitude. Furthermore,
= 3.6 x 10-1 was obtained hom the data of CM and GM, comparecl to Riesz's value
of approxhately 3.2 x IO-'.
M e n we tiinied our attention to psychophysicd adaptation, it was demonstrated
that the data of Békésy could be fitted using a value of the exponent qua1 to n =
p - q = 3.2 x IO-'. Since the data were obtained for an 800 Hz tone, we see that this
is in good agreement with Riesz's value of about 2.9 x 10-l. Furthermore, Békésy's
valiies of p = 4.1 x 10-L and q = 9.8 x IO-* are in good agreement with the eqtiation
given by the mode1 of equilibrium receptor memory in Appendix -4 (q = p / 4 , cf. Eq.
(A15)).
In the last chapter, the data of Riesz were also used to predict the magnitude
estimation data of Luce and Mo with only one additional scaling parameter.
This is perhaps only the second time in the history of the sensory sciences that a
common set of parameters has ever been found for one modality. The first attempt
was made by Norwich in 1993 for gustation using NaCl data.
Thiis, it appears that there is indeed some coherence between the parameters ob-
tained hom the different psychoacoustical experirnents. However, there stiil remains
the problem of obtaining a complete set of results for a single subject. This task must
also be repeated for the other modalities as weU.
8.3 The Threshold Associated with Simple
React ion Time
As mentioned earlier, measurements of reaction times are not u s u d y associated with
threshold phenornena. Consequently, we divided the material into two separate chap
ters (5 and 6). However, we have demonstrated that there is also a threshold as-
sociated with reaction times. bthermore, it was postulated that this threshold is
governed by the same condition as are the differential and the absolute thresholds.
The concept that reaction times and threshold phenornena are two sides of the
same coin is a relatively new idea.
8.4 Two Remaining Points.. .
( 1) The original entropy theory put forth the idea that perception can only occur when
there is a gain in information. However, we saw in Section 5.3.4 that perception can
also occiu when information is lost. This opens up an entirely new way of interpreting
the nature of perception.
(2) Two parameters have been identified representing the fundamental lower limit
to sensory ability I, represents the smdest signal which is detectable by the sensory
system. Atr represents the shortest time for which the sensory system can process
stimulus information. Both terrns are iinear functions of the information threshold,
AH.
Chapter 9
Conclusions
9.1 General Remarks
This thesis is primarily about unification. However, unification is only possible when
the imderlying conceptiial foundation has been elucidated.
The ideas of the sensory entropy theory as pioneered by Professor Kenneth Nor-
wich have been pivotal in establishing what rnay be the very first conceptual and
mathematical theory of sensory perception.
Based upon the hypothesis that entropy or uncertainty is the prirnary "state vari-
able" of perception, ure are able to dernonstrate, with entropy theory, that much if not
all of basic sensory studies can be understood in t e m of a single derived equation.
In this thesis, 1 have attempted to continue the work of Prof. Norwich by extendhg
the classical entropy theory. Having developed what I termed the universal entropy
theory as a generalization upon the classical theory, 1 then demonstrate that this
theory is capable of accoiinting for (1) almost ad of the experimentdy observed
neiuai phenornena issillng fiom the response of an isolated sensory receptor with
its associated p r i m q afFerent neuron; and (2) almost a l l of the empirical resdts
observeci in s tiidies on t hreshold and reaction tirnes.
Of particdar importance is the introduction of the threshold hypothesis and the
idea that there is a threshold governing reaction times.
9.2 Specific Remarks
In this t hesis' 1 sought to demonstrate that many of the sensory equations discovered
ernpirically issue from a single equation derived entropicdy. This equation relates the
firing freqiiency F of a primary aiferent sensory neuron to both the stimulus intensity
I and stimulus duration t. That is, F = F ( I , t ) .
F (1. t ) represents a sensory input/output "transfer function" : Given any time-
varying inpiit 1, the conesponding biological output or response F can be predicted.
Several highlights hom the thesis include:
9.2.1 Neurophysiological P henomena
The single equation, which is temeci the neural entropy equation, is capable of pre-
dic ting
(1) The neilrd adaptation effect (F vs. t) and driven neural response (F vs. I).
For example, the neural entropy equation gives a good theoretical fit to the data of
Brachman and Smith (see Smith' 1988) rneasiired from the inner hair cells of the
Mongolian Gerbil.
(2) The effects of the stimulus rise time on the neural adaptation curve as weU as
the de-adapta t ion response, incremental response and spontaneous neural ac t i v i .
Much of (1) and (2) can be found in a recent publication (Norwich and Wong,
1995).
9.2.2 Psychophysical Phenornena
The same equation can also account for psychophysical phenomena such as
(1) Psychoacoiistical adaptation as measured by Békésy (1929), rnatching the loiid-
ness of two tones of dinerent durationsr
(2) Magnitude estimation data of, say, Luce and Mo (1965).
Furthermore, when combineci with a threshold hypothesis, this equation can be
iised to derive
(3) The Weber fraction equation of Knudsen (1923) and Riesz (1928);
(4) The curves of Garner and bUer (1947) and Roufs (1971) measuring Weber
fraction vs. increment duration;
(5) The absoliite threshold equation of Plomp and Bouman (1959), Zwislocki
(1960);
(6) Miller's observation (1947) that the slope of the Weber fraction cuwe must fall
Mth slope -1 near threshold on a log(AI/I) W. log(I) plot;
(7) Bloch's law and Garner's (1947) modification to B1och.s law:
(8) Piéron's law for simple reaction times.
The theory also identifies a new parameter 61 which represents a signal interna1
to the sensory organ. 6 1 determines the absolute threshold, Ithrhresh, and Ithrerh oc 61.
For example, Zwislocki (1965) postulated the that 61 2.5 Ithresh. This resd t has
now been derived in my thesis.
Some of the resdts mentioned above have b e n published in Wong and Norwich
(1995), Xorwich and Wong (1996) and Wong and Norwich (1996).
A guide to, and siimmary of, the developments in this thesis is shown in Fig. 25.
A Guide to This Thesis
Introduction to Sensory Entropy
Theory
C lass ical Equation Section 3.1
Chapter 3
4
/ Derivation of the ) Universal Equation
Phenomena
1 Quantum Threshold 1
Application to Direct Psychophysical
Response Chopter 7
1 Appiicationto Sensory Thresholds
Chapter 5
Hypothesis 1 \
Application to Simple Reaction
Time Chopter 6
Figure 25: A guide to this thesis.
Appendix A
A Model of Equilibrium Recep-
tor Memory
In this appendix a simple mode1 is developed which can account for the power de-
pendence of me, on I (see Eq. (38)). Actudy, it suffices to show that me, a Iq since
we can treat the tenn I + S I as a combined intensity.
A.1 Physical Model
In the top diagram of Fig. 26, we see an explicit time sequence of receptor sam-
ples. Each cross represents one samphg, and the sample size rn increases with each
interval. We shall suppose that each sampling represents a measurement made by,
Say. an olfactory receptor. These rneasurements would reflect the density of odorant
molecules in a local receptor "subvolurne" . Thus, a typical sequence of sarnples might
be similar to the one found in the lower diagram of Fig. 26.
These numbers were drawn from a Poisson distribution (assuming the odorant
molecules follow ideal gas statistics) with mean = 4. I have circled all instances of
the number '3' for later use.
It is assumed that the receptor takes its e s t sampling as a "standard". Al1 future
Figure 26: A time sequence of receptor samples. Each value represents one samphg.
In this case, the receptor is counting the number of times the value '3' is measured.
sarnplings are to be compared to this one standard. In the example shown in Fig. 26.
the density value of '3' is taken as the standard. We now postdate that the receptor
looks for a repetitive pattern in the density sarnpling sequence. That is, the receptor
is searching for repetitions of the standard value (in this case '3') in its sampling
seqiience.
How many repetitions might the receptor look for? Let ris Say that the receptor
coimts d occurrences of '3' during the course of adaptation. n, can then be defined
as the number of samples required to count 4 occurrences of the standard '3' for a
particular mean density (in this case mean = 4). When rn = me, it is assumed that
the receptor no longer increases its sample size.
Continuing our example, we see that if 4 = 3 then me, = 4. There are 3 occurrences
of the value '3' by the fourth sample (in total there are 4), and thus me, takes on the
value of 4.
One might also nrish to refine the definition of me, by averaging it over all possible
initial values or standards so as to produce an average value of m.,. We s h d proceed
by this method.
Derivat ion of the
Receptor Memory
Power Dependence
Let r i s now look at the problem in the abstract.
Denoting i as the density value obtained in the first sample, i becomes the value of
the standard. The probability of sampling a value i is given by the Poisson distribution
p ji: A) . where X is the mean density:
For the Nth sample, there will be N samplings drawn £rom the population. On
average, t here wilI be N p (2; A) occurrences of i. Of course, one must remember that
N p (2; A) is an integer. This condition is satisfied if N is large.
\?le require that after m = me, samples, the counted number of occurrences of 2 is
Since p ( 2 ; A) remains constant in the sum, we obtain
m c ~
p (i; A) N = p ( 2 ; A) N = l
Solving for me, gives
-1 + 41 + 8 # / p (2; A) meq = 2
Since me, 2 O, we have discarded the negative root. U'e can now average me, over
d l possible initial conditions,
To demonstrate that Eq. (A5) gives a power dependence of . f ieq on the mean
density, A, we must make several approximations.
CVriting out Eq. (A5) in fidl we have
In general? 8 $ / p (2: A) >> 1 because C$ 2 1 and p (i; A) < 1 by definition. Therefore.
\Ne s h d now adopt the large X approximation so that the Poisson distribution can
be approximated by the normal distribution with &ance equal to mean. Thus,
Since the step size is small compared to the fùll range of summation, we can approx-
irnate the sum by an integral,
Using the result
we obtain
Since both X and o w-iU tend to be large, we can take erf (9) = 1 and drop the b a l
term of 1 to obtain finally.
which is the power dependence we were looking for when we take 1 = A.
This eqtiation can be cornpareci to vdties obtained by simulation. In Fig. 27, the
simulation restdts are plotted.
Taking the logarithm of both sides in Eq. (A12), we obtain
Siibstitiiting the value of d = 50 into t h s equation, we obtain a slope of 0.25 and
an intercept of 3.1. The best-fit line in Fig. 27 has slope 2.0 x IO-' & 3 x 1 0 - ~ and
intercept 3.1 & 1 x 10-l.
Eq. (A12) constrains the value of q to be 0.25. However, we do not have to restrict
oiirselves to a Poisson distribution - a general normal distribution can be used in its
place. In this case, variance does not necessarily qua1 mean.
However, if the variance is related to the mean in the form
(cf. Eq. (36)), repeating the above cdculations for the normal distribution would
yield
Figure 27: Simulation of the equilibrium receptor memory mode1 with Q = 50. The
bat-fit straight line has approximately the same dope and intercept as predicted by
Eq. (A12).
In this case. we have q = p / 4 .
Appendix B
Deriving the Neural Response to a
Double Step Input
GVe can easily solve for the neural response found in Fig. 17 given the intensity profile
in Fig 16.
The response in region 1 is given simply by Eq. (44) with the time origin shifted
from t = O to t = -7.
To obtain the response in region 2, we first assume that T is of sufbcient length that
the receptor has completely adapted to the pedestal. Thus, m (O) = m~,Ïq, where
Ï = 1 + 61, and, referring to Eqs. (38), (40) and (43), we have
rn ( t ) = Ï q ë a t + (Ï+ AI)' ( 1 - ëat)
= Ï q [e-at + (1 + (1 - ëat)] ,
where Y = A I / Ï and me, = meq (Ï + ~ 1 ) ~ .
Substitiiting m ( t ) into Eq. (37), we obtain for the receptor entropy in region 2
where 3 = p/rnzq and C = pÏn = B7p-q. Once again, we have set 0' = mz,
Appendix C
Deriving the Differential Thresh-
old Equation for the Continuous
Incrernent
In this appendix, we derive Eq. (52) from Eqs. (50) and (51).
Using Eq. (5 1) , we can evaluate the difference in H between [t,, At] to obtain
where we have iised the approximation t, = O from t , < At. This difference must
exceed AH for detectability. Equating the two quantities, we obtain
Here we make the crucial assumption that Y = AI/ (1 + 61) is s m d . R e c d that
61 is greater than A I ( s e Section 8.1). In general, even as I -r O, Y « 1 is still
not a bad approximation because A I < 61. The validity of these approximations will
become clearer as we proceed. Ussing Y « 1, tve can expand the right hand side of
Eq. (C3) to first order in Y to obtain
The approximation of Eq. (C3) by Eq. (C4) may be verified manuah , or by entering
Eq. (C3) as an eaupression in a symbolic manipulator (eg. Maple) and requesting a
first-order Taylor series approximation. We have included a Maple worksheet detailing
this calculation in Appendix F.
If A H is also s m d , using e2PH = 1 + 2 A H 1 Rie codd then mite, approximately,
or.
which is Eq. (52).
Appendix D
Deriving Eq. (59)
Once again we evaluate the difference in H between [t,, At] iising Eq. (5 l), to obtain
where < = 3AIn, x = ( 6 1 / ~ l I ) ~ and x = e-"'~. Solving Eq. (Dl ) for e-aAt gives r 1
Recall that q < p since n = p - q is always positive hom our experience. Ehrther-
more. O < q < 1 (see Section 3.2.1) and xl/q is of power greater than 1. Solving Eq.
(D2) for At and taking a first order expansion in x, we obtain
A fiirther expansion to fust order in A H yields
Siibstituting for X , C and x, we obtain, findy,
which is Eq. (59).
A Mapie worksheet d e t a h g this calcrilation is shown in Appendix F.
Appendix E
The Absolute Threshold of Hu-
man Hearing
The absoliite threshold Ithreoh for human hearing has b e n measured by many inves
tigators. In Fig. 28, the data of Wegel (1932) are shown. Notice that the threshold
is a fimction of soimd frequency.
We fitted his data with the empirical equation
2 11.43
10 log,, (-) = 17.37' [iog,, (A)] +3-468 x 1oe6 [log,, (f )] - 1 (El)
where Io is the standard intensity 10-l2 W/m2. The data and fitted curve are shonm
in Fig. 28.
Figure 28: The absolute threshold measured for Merent sound frequencies. Data of
Wegel (1932). The data were fitted by the empirical Eq. (El).
Appendix F
Maple Worksheets
On the following pages, the Maple worksheets detailhg the calculations in Appendix
C and D are shown.
Page 126 shows the derivation of the dinerential threshold Eq. (C5) (Appendix C),
and pages 127 and 128 show the derivation of the reaction time Eq. (D5) (Appendix
D)-
For convenience, the following parameters were renamed: AH = DH and At = Dt.
All other parameters and variables remained unchangeci.
Maple V Release 3 1
> simplify (" ) ;
Maple V Release 3 1
> simplify (sube(O=O,"));
Mapie V Release 3 2
Appendix G
Ali fitted cimes were obtained using a computer program implernenting the Lbsirnplex
algorithm" proposed by Nelder and Mead (1965). This method minimizes the sum-of-
square-error (SSE) between the measured data and the theoretical curve. The squate
error is calculated by taking the difference between the fitted value, yth,,, calculated
ith pair of 1V measured points, the sum-of-square-error is dehed as
in SSE is achieved.
The simplex process can be generalized to any number of theoretical c w e s and
sets of measured data. A cornbined SSE can be obtained by adding together the
individual SSE's. A common set of parameters can be obtained by minimizing the
combined sum-of-squareenor.
Sensitivity analysis is an integral part of determining the robustness of fitted pa-
rameters. Ideally, the SSE shows a strong global minimum with respect to one set
of parameters. In other words, a small change in parameter values produces a large
change in SSE. However, in reality, both the scatter in the data as well as the corn-
plexity of the fitted eqiiation can give rise to an error space which contains many
local minima. Since the simplex algorithm can guarantee only hding a local (and
not global) minimum, how can we h u r e that the fitted parameters give the lowest
SSE? Rirthermore. if several minima show approximately the same sum-of-square
error, which set of parameters should be chosen as representative?
PVhile the theory of sençitivi ty analysis has b e n developed quite extensively for
dynamical systems and is relatively easy to use (e-g. see Hearne, 1985), sensitivity
analysis for more complicated equations or systems of equations require much greater
effort (Press et al, 1986).
One method to evaluate the robiistness of parameters is by repeating the cime-
fit ting procedure sever al times with significantly different starting points for each
parameter value. If approximatelq. the same final values are obtained each time then
we can be reasonably sure these values characterize a global minimum. This is the
method adopted for this thesis.
To take an example, the data of Stevens (1969) from Fig. 11 were fitted 6 times
in log-log coordinates with initial parameter values differing by at least one order of
magnitude. The results show that the b a l parameter values never ClifFer by more
than 4 parts in IO4:
Initial valiies I Final values
The second part of the analysis involves looking at how a s m d change in parameter
value will infliience the sum-of-square-error. Summarized in the following table are
the residts of changing the values of the parameters, one at a time, by f 10% fkom t heir
final values and looking at the corresponding change in SSE. The values k = 41.5604,
0 = 0.0985774, n = 1.488741 and SSE = 0.3099874 were chosen as representative
133
k
0.1
0.1
1
1
10
10
I
SSE
0.3099874
0.3099875
0.3099874
0.3099874
0.3099875
0.30998'75
3
1
n
1.488741
1.488846
1.488829
1.488873
1.488868
1.488859
k
41.5604
41.54814
41.54536
41.54431
41.54334
41.55325
n
10
P
0.0985774
0.0986096
0.0986133
0.0986163
0.0986183
0.0985915
10
0.1
10
0.1
1
1
10
0.1
1
0.1
final values:
From this table, we see that a 10% change in parameter values produces at Ieast a
15% change in SSE. This is a good indication that the fitted values are robust.
k + 10%
k - 10%
$3 + 10%
,3 - 10%
n t 10%
n, - 10%
k
45.71644
37.40436
% change in SSE
26%
32%
,Li'
-
-
15%
18%
54%
63%
0-3555062
0.3668767
0.4760579
n
-
-
SSE
0.3917878
0.4098463
-
-
1.637615 1
- 1 0.108435
-
-
1.3398669 1 0.5062194
0.0887195
-
- -
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